yC-NRLF. 7M7 RUDIMENTS A? T ONT NHMEROUS EXERCISES SLATE AND BTACKBOARD, FOR BI1-- BY OAMES B. iilOjViSON, A.M., AUTHOR OF >*' , EXERCISKS IN ARITn!ft<CTICAIi ANfALYS PRACTICAL ARITTIMKTIC ; UIGHKR ARITHMETIC; EDITOR OF DAV's SCHOOL AL^KBK/t, '.KG^NDRE's GEOMSTRV> ETC, oo CM O ,W YOKE: N , c \ T EY, 48 & 50 WALKER ST. AGO : S. C. G RIGOS & CO., 89 & 41 LAK T :vs & co. ST. LOUIS : KEITH & WOODS. ,. x . -23 & CO. DETROIT I EAYMOIO & SKLLiiOK. -.i.riMS. BTJFFALO: PH1NNKY A CO. .vaUSfr: T. S. Q[JAOKENBU8H.
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yC-NRLF.
7M7
RUDIMENTS
A? T
ONT
NHMEROUS EXERCISES
SLATE AND BTACKBOARD,
FOR BI1--
BY OAMES B. iilOjViSON, A.M.,AUTHOR OF >*'
, EXERCISKS IN ARITn!ft<CTICAIi ANfALYS
PRACTICAL ARITTIMKTIC ; UIGHKR ARITHMETIC; EDITOR OF DAV's
SCHOOL AL^KBK/t, '.KG^NDRE's GEOMSTRV> ETC,
oo
CMO,W YOKE:
N ,
c \T
EY, 48 & 50 WALKER ST.AGO : S. C. G RIGOS & CO., 89 & 41 LAKT
:vs & co. ST. LOUIS : KEITH & WOODS.,.x
. -23 & CO. DETROIT I EAYMOIO & SKLLiiOK.-.i.riMS. BTJFFALO: PH1NNKY A CO.
.vaUSfr: T. S. Q[JAOKENBU8H.
.
LIBRARYOF THE
UNIVERSITY OF CALIFOFGIFT OF
Received....
Accession No. 6 J ^ ^ 3~ - Class No.
an& ffifjomsan's Series.
RUDIMENTSOF
AEITHMETIC;CONTAINING
NUMEROUS EXERCISES
SI-ATE AND S&ACKBOARD,
FOB BEGINNERS
BY JAMES IB. THOMSON, A.M.,JSFf P* fllf Tjfl B ^x^^1^^
AUTHOR OK MENTAL ARiTHMETicr; EXERCISES IN ARIT&MKTICAL ANALYST**
PRACTICAL ARITHMETIC; HtaH^fa ARITHMETIC; E|<ITOROP DA\ s
SCHOOL ALGEBRA; LEGENT>RK ySGEOMETR.V, ETC.
NEW TOEK:IVISON & PHINNEY, 48 & 50 WALKER ST.CHICAGO: S. C. GEIGGS & CO., 39 & 41 LAKE ST.
CINCINNATI : MOORE, WIL8TACH, KEYS & CO. ST. LOUIS : KEITH & WOODS.PHILADELPHIA; SOWER, BARNES & oo. BUFFALO; PHINNEY fc CO.
NEWBUEQ: T. s. QTTAOKENBUSH. t
1859.
o
Entered according to Act of Congress, in the year 1833, Ifjr
JAMES B. THOMSON,
fa the Clerk's Office for the Southern District of New Yoft
TKREOTTPXD BY THOMAS B. 81CITK,216 W1LLIAH 8TRKKT, N. V.
PREFACE.
EDUCATION, in its comprehensive sense, is the business of
life. The exercises of the school-room lay the foundation;
the superstructure is the work of after years. If these exer-
cises are rightly conducted, the pupil ohtains the rudiments
of science, and what is more important, he learns how to
study ,how to think and reason, and is thus enabled to appro-
priate the means of knowledge to his future advancement.
Any system of instruction, therefore, which does not embrace
these objects, which treats a child as a mere passive recipient,
is palpably defective. It is destitute of some of the most
essential means of mental development, and is calculated to
produce pigmies, instead of giant intellects.
The question is often asked," What is the best method of
proceeding with pupils commencing the study of Arithmetic,
or entering upon a new rule ?"
The old method. Some teachers allow every pupil to cipheru on his own hook;" to go as fast, or as slow as he pleases,
without reciting a single example or rule, or stopping to in-
quire the "why and the wherefore" of a single operation.
This mode of teaching is a relic of by-gone days, and is primafade evidence, that those who practice it, are behind the
spirit of the times.
Another method. Others who admit the necessity of teach-
ing arithmetic in classes, send their pupils to their seats, and
tell them to ustudy the rule." After idling away an hour
or more, up goes one little hand after another with the de-
spairing question :
" Please to show me how to do this sum,sir ?" The teacher replies,
"Study the rule
;that will tell
you." At length, to silence their increasing importunity, he
takes the slate, solves the question, and, without a word of
5V PREFACE,
explanation, returns it to its owner. He thus goes through
the class. When the hour of recitation comes, the class is
not prepared with the lesson. They are sent to their seats
to make another trial, which results in no "better success.
And what is the consequence? They are discouraged and
disgusted with the study.A more excellent way. Other teachers pursue a more ex-
cellent way, especially for young pupils. It is this : The
teacher reads over with the class the preliminary explanations,
and after satisfying himself that they understand the mean-
ing of the terms, he calls upon one to read and analyze the
first example, and set it down upon the blackboard, while
the rest write it upon their slates. The one at the bourd
then performs the operation audibly, and those with their
slates follow step by step.
Another is now called to the board and requested to set
down the second example, while the rest write the same
upon their slates, and solve it in a similar manner. He then
directs them to take the third example, and lets them try
their own skill, giving each such aid as he may require. In
this way they soon get hold of the principle, and if now sent
to their seats, will master the lesson with positive delight.
As to assistance, no specific directions can he given whichwill meet every case. The best rule is, to afford the
learner just that kind and amount, which will secure the
greatest degree of exertion on his part. Less than this dis-
courages; more, enervates.
In conclusion, we would add, that this elementary workwas undertaken at the particular request of several eminent
practical teachers, and is designed to fill a niche in primaryschools. It presents, in a cheap form, a series of progressiveexercises in the simple and compound rules, which are
adapted to the capacities of beginners, and are calculated to
form habits of study, awaken the attention, and strengthenthe intellect.
J. B. THOMSON.KBW YORK, January, 1858.
CONTENTS
SECTION I.
ARITHMETIC defined, ....--7Notation, 7
Roman Notation, ......--7Arabic Notation. ...... -9Numeration, -12
SECTION II.
ADDITION defined, 16
When the sum of a column does not exceed 9,- - -18
When the sura of a column exceeds 9,- - - - -19General Rule for Addition, - - - - - - -20
SECTION III.
SUBTRACTION defined, 27
When a figure in the lower No. is smaller than that above it,- 28
When a figure in the lower No. is larger than that above it, 29
Borrowing 10, .... 30
General Rule for Subtraction, - - - .. -81SECTION IV.
MULTIPLICATION defined, 36
When the multiplier contains but one figure, 89
When the multiplier contains more than one figure,- - 41
General Rule for Multiplication,.... 43
To multiply by 10, 100, 1000, <fcc., 45
When there are ciphers on the right of the multiplier,- 46
When there are ciphers on the right of the multiplicand,- 47
When there are ciphers on the right of both, 48
SECTION V.
DIVISION defined, ....... . . 49
Short Division, - 52
VI CONTENTS.
Rule for Short Di nsion, 54
Long Division, 56Difference between Short and Long Division, - - - 67Rule for Long Division, 58To dhide by 10, 100, 1000, <fec.,
- 61
When there are ciphers on the right of the divisor,- - 62
SECTION VI.
FRACTIONS, - 68
To find what part one given number is of another, - - 66
A part of a number being given, to find the whole, - - 66
To multiply a whole number by a fraction, - - - 67
To multiply a whole by a mixed number, - - - - 69
To divide a whole number by a fraction, 70To divide a whole by a mixed number, - - - -71
SECTION VII.
TABLES in Compound Numbers, - - - 74
Paper and Books, - - .. . . . -85Tables of aliquot parts,
------- 87
SECTION VIII.
ADDITION of Federal Money, 90
Subtraction of Federal Money, 92
Multiplication of Federal Money, 93
Division of Federal Money, 94
SECTION IX.
REDUCTION, 96
Rule for Reduction Descending, 97
Rule for Reduction Ascending, 100
Compound Addition, 106
Compound Subtraction, 108
Compound Multiplication, 110
Compound Division,- - - - - - - -11.1
Miscellaneous Exercises, - * - - - 113
Answers to Examples,- - - 119
ARITHMETIC.
SECTION I.
ART, ! ARITHMETIC is the science of numbers.
Any single thing, as a peach, a rose, a book, is called a
unit, or one ; if another single thing is put with it, the
collection is called two ; if another still, it is called three ;
if another, four ; if another, five, &c.
The terms, one, tioo, three, four, <kc., are tke names of
numbers. Hence,
2. NUMBER signifies a unit, or a collection of units.
Numbers are expressed by words, by letters, and bj
figures.
3* NOTATION is the art of expressing numbers by letters
or figures. There are two methods of notation in use, the
Roman and the Arabic.
I. ROMAN NOTATION.
4r The Roman Notation is the method of expressingnumbers by letters ; and is so called because it was invented
by the ancient Romans. It employs seven capital letters,
viz : I, V, X, L, C, D, M.
When standing alone, the letter I, denotes one ; V, fiv-e ;
X, ten ; L, fifty ; C, one hundred ; D, five hundred ; M,one thousand.
QUEST. 1. What is Arithmetic ? What is a single thing called? If an-
other is put with it, what is the collection called? If another, what ? Whatare the terras one, two, three, &c. ? 2. What then is number ? How are
numbers expressed ? 3. What is Notation ? How many methods ofnotation
are in use? 4. What is the Roman notation? Why so called? How manyetters does it employ? What does the letter I, denote? V? X? L? C? D? M?
NOTATION. !SECT. i.
5. To express the intervening numbers from to one a
thousand, or any number larger than a thousand, we re*
sort to repetitions and various combinations of these let-
ters, as may be seen from the following
TABLE.I denotes
ARTS. 5 7.] NOTATION. 9
OBS. 1. Every time a letter is repeated, its value is repeated.
Thus, the letter I, standing alone, denotes one ; II, two ones or two,
<fcc. So X denotes ten ; XX, twenty, <fcc.
2. When two letters of different value are joined together, if the
less is placed before the greater, the value of the greater is dimin-
ished as many units as the less denotes;if placed after the greater,
the value of the greater is increased as many units as the less de-
notes. Thus, V denotes five;but IV denotes only four
;and VI,
six. So X denotes ten; IX, nine
; XI, eleven.
Note. The questions on the observations may be omitted, by
beginners, till review, if deemed advisable by the teacher.
II. AKABIC NOTATION.
6. The Arabic Notation is the method of expressing
numbers by figures ; and is so called because it is supposedto have been invented by the Arabs. It employs the fol-
lowing ten characters or figures, viz :1234567890one, two, three, four, five, six, seven, eight, nine, naught.
OBS. 1. The first nine are called significant figures, because each
one always expresses a value, or denotes some number. They are
also called digits, from the Latin word digitus, signifying a finger,
because the ancients used to count on their fingers.
2. The last one is called naught, because when standing alone,
it expresses nothing, or the absence of number. It is also called
cipher or zero.
7 All numbers larger than 9, are expressed by different
combinations of these ten figures. For example, to express
ten, we use the 1 and 0, thus 10;to express eleven, we
use two Is, thus 11;to express twelve, we use the 1 and
2, thus 12, <fec.
QuKST.Oi*. What is the effect of repeating a letter ? If a letter of lost
value is placed before another of greater value, what is the effect? If placed
after, what ? 6. What is the Arabic notation ? Why so called ? How manyfigures does it employ? What are their names? Obs. What are the first nino
called ? Why ? What else are they sometimes called? What is the last onecalled? Why? 7. How are numbers larger than nine expressed ? Expressten by figures. Eleven. Twelve. Fifteen.
10 NOTATION. [SECT, i
ARTS. 8 11.] NOTATION. 11
8. It will be peiceived from the foregoing table, that
the same figures, standing in different places, have differ-
ent values.
When they stand alone or in the right hand place, they
express units or ones, which are called units of the first
order.
When they stand in the second place, they express tens,
which are called units of the second order.
When they stand in the third place, they express hun-
dreds, which are called units of the third order.
When they stand in the fourth place, they express
thousands, which are called units of the fourth order, <fec.
For example, the figures 2, 3, 4, and 5, when arranged
units;when arranged thus, 5432, they denote 5 thousands,
4 hundreds, 3 tens, and 2 units.
9 Ten units make one ten, ten tens make one hundred,and ten hundreds make one thousand, &c.
;that is, ten of
any lower order, are equal to one in the next higher order
Hence, universally,
O. Numbers increase from right to left in a tenfold
ratio ; that is, each removal of a figure one place towards
the left, increases its value ten times.
11. The different values which the same figures have,are called simple and local values.
The simple value of a figure is the value which it ex-
presses when it stands alone, or in the right hand place.
QUEST. 8. Do the same figures always have the same value ? When stand-
ing alone or in the right hand place, what do they express? What do they
express when standing in the second place? In the third place? In the
fourth ? 9. How many units make one ten ? How many tens make a hun-
dred ? How many hundreds make a thousand ? Generally, how many of anylower order are required to make one of the next higher order ? 10. What is
the general law by which numbers increase ? What is the effect upon the value
of a figure to remove it one place towards the left? 11. What are the differ*
ent values of the same figure called ? What is the simple value of a figure ?
What the local value ?
12 NUMERATION. [SECT. 1.
The simple value of a figure, therefore, is the numbelwhich its name denotes.
The local value of a figure is the increased value which:t expresses by having other figures placed on its right.
Hence, the local value of a figure depends on its locality,or the place which it occupies in relation to other num-bers with which it is connected. (Art. 8.)
OBS. This system of notation is also called the decimal system,because numbers increase in a tenfold ratio. The term decimal if
derived from the Latin word decem, which signifies ten.
NUMERATION.
12* The art of reading numbers when expressed by
figures, is called Numeration.
NUMERATION TABLE.
123 861 518 924 263
Period V. Period IV. Period III. Period II. Period I.
Trillions. Billions. Millions. Thousands. Units.
13. The different orders of numbers are divided into
periods of three figures each, 'beginning at the right hand.
QUEST. Upon what does the local value of a figure depend ? Obs. What ia
this system of notation sometimes called ? Why ? 12. What is Numeration ?
Repeat the numeration table, beginning at the right hand. What is the first
place on the right called? The second place? The third? Fourth? Fifth
Sixth? Seventh? Eighth? Ninth? Tenth, &c.? 13. How are the orders of
numbers divided ?
ARTS. 12 14.] NUMERATION. 13
The first, or right hand period is occupied by units, tens,
hundreds, and is called units' period ;the second is oc-
cupied by thousands, tens of thousands, hundreds of
thousands, and is called thousands' period, &c.
The figures in the table are read thus : One hundred
and twenty-three trillions, eight hundred and sixty-one
billions, five hundred and eighteen millions, nine hundred
and twenty-four thousand, two hundred and sixty-three.
1 4 To read numbers which are expressed by figures.
Point them off into periods of three figures each ; then,
beginning at the left hand, read the figures of each period
as though it stood alone, and to the last figure of each, add
the name of the period.
OBS. 1. The learner must be careful, in pointing 0$~ figures, alwaysto begin at the right hand ;
and in reading them, to be^in at the
left hand.
2. Since the figures in the first or right hand period alw lys de-
note units, the name of the period is not pronounced. Hevxce, in
reading figures, when no period is mentioned, it is always u,ider-
stood to be the right hand, or units' period.
EXERCISES IN NUMERATION.
Note. At first the pupil should be required to apply to each fig-
ure the name of the place which it occupies. Thus, beginning at
the right hand, he should say,"Units, tens, hundreds," &c., and
point at the same time to the figure standing in the place which he
mentions. It will be a profitable exercise for young scholars to
write the examples upon their slates or paper, then point them off
into periods, and read them.
QUEST. What is the first period called ? By what is it occupied ? What is
the second period called? By what occupied? What is the third periodcalled ? By what occupied ? Wnat is the fourth called ? By what occupied f
What is the fifth called? By what occupied? 14. How do you read nurabere
expressed by figures ? Obs. Where begin to point them off? Where to read
them ? Do you pronounce the name of the right hand periol ? When no
period is named, what is understood ?
14 NUMERATION.
Read the following numbers :
[SECT. I
Ex. 1.
ART. 15.J NUMERATION. 19
5. One hundred and ten.
6. Two hundred and thirty-five.
7. Three hundred and sixty.
8. Two hundred and seven.
9. Four hundred and eighty-one.
10. Six hundred and ninety-seven.
11. One thousand, two hundred and sixty-three.
12. Four thousand, seven hundred and ninety-nine.
13. Sixty-five thousand and three hundred.
14. One hundred and twelve thousand, six hundred
and seventy-three.
15. Three hundred and forty thousand, four hundred
and eighty-five.
16. Two millions, five hundred and sixty thousand.
17. Eight millions, two hundred and five thousand,
three hundred and forty-five.
18. Ten millions, five hundred thousand, six hundred
and ninety-five.
19. Seventeen millions, six hundred and forty-five
thousand, two hundred and six.
20. Forty-one millions, six hundred and twenty thou-
sand, one hundred and twenty-six.
21. Twenty-two millions, six hundred thousand, one
hundred and forty-seven.
22. Three hundred and sixty millions, nine hundred
and fifty thousand, two hundred and seventy.
23. Five billions, six hundred and twenty-one millions,
seven hundred and forty-seven thousand, nine hundred
and fifty-four.
24. Thirty-seven trillions, four hundred and sixty-three
billions, two hundred and ninety-four thousand, fire hun-
dred and seventy-two.
1$ ADDITION. [SECT. IL
SECTION II.
ADDITION.
ART. 16. Ex. 1. Henry paid 4 shillings for a pair of
gloves, 7 shillings for a cap, and 2 shillings for a knife :
how many shillings did he pay for all ?
Solution. 4 shillings and 7 shillings are 11 shillings,
and 2 shillings are 13 shillings. He therefore paid 13
shillings for all.
OBS. The preceding operation consists in finding a single num-
ber which is equal to the several given numbers united together,
and is called Addition. Hence,
17 ADDITION is the process of uniting two or more
numbers in one sum.
The answer, or number obtained by addition, is called
the sum or amount.
OBS. When the numbers to be added are all of the same kind, or
denomination, the operation is called Simple Addition.
18 Sign of Addition (+). The sign cf addition is
a perpendicular cross(+ ), called plus, and shows that
the numbers between which it is placed, are to be added
together. Thus, the expression 6 + 8, signifies that 6 is
to be added to 8. It is read," 6 plus 8," or
<: 6 added to 8."
Note. The term plus, is a Latin word, orig'nally signifying'*rcore." In Arithmetic, it means " added to."
QUEST. 17. What is addition? What is the answer called ? Obs. Whenthe numbers to bo added are all o? the same denomination, what is the ope-ration called? 18. What is the si^n of addition? Who* does it show ? JVWe.
What is the moaning of the word plus ?
ARTS. 16 19.] ADDITION. 17
19. Sign of Equality ( ).The sign of equality is
two horizontal lines( ),
and shows that the numbers be-
tween which it is placed, are equal to each other. Thus,
the expression 4+ 3= 7, denotes that 4 added to 3 are
equal to 7. It is read," 4 plus 3 equal 7," or
" the sum
cf 4 plus 3 is equal to 7." 18 + 5= 7
ADDITION TABLE.
2 and
18 ADDITION. [SECT. II.
' tab es distinctly and indelibly fixed in his mind. Hence,after a taole has been repeated by the class in concert, or individ-
ually, the teacher should ask many promiscuous questions, to preventits being recited mechanically, from a knowledge of the regular in-
crease of numbers.
EXAMPLES.
!3O When the sum of a column does not exceed 9.
Ex. 1. George gave 37 cents for his Arithmetic, and
42 cents for his Reader : how many cents did he give for
both?
Directions. Write the numbers Operation.
under each other, so that units ^ &
may stand under units, tens under jf gtens, and draw a line beneath them. 3 7 price of Arith.
Then, beginning at the right hand 4 2 " of Read.or units, add each column sepa- -rately in the following manner : 7 9 Ans.
2 units and 7 units are 9 units. Write the 9 in units
place under the column added. 4 tens and 3 tens are
Y tens. Write the 7 in tens' place. The amount is 79
cents.
Write the following examples upon the slate or black-
board, and find the sum of each in a similar manner :
(2.) (3.) (4.) (5.)
26 231 623 5734
42 358 145 4253
(6.) (7.) (8.) (9.)
425 3021 5120 3521
132 1604 2403 1043
321 2142 1375 4215
10. What is the sum of 4321 and 2475 ?
11. What is the sum of 32562 and 56214?
12. What is the sum of 521063 and 465725 ?
ARTS. 20 22.J ADDITION. 19
21. When the sum of a column exceeds 9.
13. A merchant sold a quantity of flour for 458 dollars,
a quantity of tea for 887 dollars, and sugar for 689 dol-
lars : how much did he receive for all ?
Having written the numbers as Operation.
Defore, we proceed thus: 9 units 458 price of flour,
and 7 units are 16 units, and 8 887 " of tea.
are 24 units, or we may simply 689 " of sugar,
say 9 and 7 are 16, and 8 are 24. 2034 dollars. Ans.
Now 24 is equal to 2 tens and
4 units. We therefore set the 4 units or right hand figure
in units' place, because they are units ; and reserving the
2 tens or left hand figure in the mind, add it to the column
of tens because it is tens. Thus, 2 (which was reserved)
and 8 are 10, and 8 are 18, and 5 are 23. Set the 3 or
right hand figure under the column added, and reserving
the 2 or left hand figure in the mind, add it to the column
of hundreds, because it is hundreds. Thus, 2 (which was
reserved) and 6 are 8, and 8 are 16, and 4 are 20. Set
the or right hand figure under the column added ; and
since there is no other column to be added, write the 2
in thousands' place, because it is thousands.
N. B. The pupil must remember, in all cases, to set down the
whole sum of the last or l$ft hand column.
22. The process of reserving the tens or left hand fig-
ure, when the sum of a column exceeds 9, and adding it
mentally to the next column, is called carrying tens.
Find the sum of each of the following examples in a
similar manner :
(14.) (15.) (16.) (17.)
856 364 6502 8245
764 488 497 4678
1620 Ans. 602 8301 362
20 ADDITION. [SECT. IL
23. From the preceding illustrations and principles
we derive the following
GENERAL RULE FOR ADDITION".
I. Write the numbers to be added under each other, so
that units may stand under units, tens under tens, &c.
II. Beginning at the right hand, add each column sepa-
rately, and if the sum of a column does not exceed 9, write
it under the column added. But if the sum of a column
exceeds 9, write the units' figure under the column and
carry the tens to the next column.
III. Proceed in this manner through all the orders, and
finally set down the whole sum of the last or left hand
column.
24. PROOF. Beginning at the top, add each column
downward, and if the second result is the same as the
first, the work is supposed to be right t
EXAMPLES FOR PRACTICE.
(1.) (2.) (3.) (4.)Pounds. Feet. Dollars Yards.
25 113 342 4608
46 84 720 635
_84 2_16898 43
(5.) (6.) (7.) (8.)
684 336 6387 8261
948 859 593 387
569 698 3045 13
203 872 15 7^
9. What is the sum of 46 inches and 38 inches?
QUEST. 23. How do you write numbers for addition? When tho mim of a
column does not exceed 9, how proceed ? When it exceeds 9, how proceed ?
22. What is meant by carrying the tens ? What do you do with the sum of
the last column ? 24. How is addition proved ?
< 23, 24.] ADDITION. 21
10. What is the sum of 51 feet and 63 feet ?
11. What is the sum of 75 dollars and 93 dollars?%
12. Add together 45 rods, 63 rods, and 84 rods.
13. Add together 125 pounds, 231 pounds, 426 pounds.
14. Add together 267 yards, 488 yards, and 625 yards.
15. Henry traveled 256 miles by steamboat and 320
miles by Railroad : how many miles did he travel ?
16. George met two droves of sheep ; one contained
461, and the other 375 : how many sheep were there in
both droves ?
17. If I pay 230 dollars for a horse, and 385 dollars for
a chaise, how much shall I pay for both ?
18. A farmer paid 85 dollars for a yoke of oxen, 27
dollars for a cow, and 69 dollars for a horse : how muchdid he pay for all ?
19. Find the sum of 425, 346, and 681.
20. Find the sum of 135, 342, and 778.
21. Find the sum of 460, 845, and 576.
22. Find the sum of 2345, 4088, and 260.
23. Find the sum of 8990, 5632, and 5863.
24. Find the sum of 2842, 6361, and 523.
25. Find the sum of 602, 173, 586, and 408.
26. Find the sum of 424, 375, 626, and 75.
27. Find the sum of 24367, 61545, and 20372.
28. Find the sum of 43200, 72134, and 56324.
29. A young man paid 5 dollars for a hat; 6 dollars
for a pair of boots, 27 dollars for a suit of clothes, and 19
dollars for a cloak : how much did he pay for all ?
30. A man paid 14 dollars for wood, 16 dollars for a
stove, and 28 dollars for coal : how many dollars did he
pay for all ?
31. A farmer bought a plough for 13 dollars, a cart
for 46 dollars, and a wagon for 61 dollars : what was the
price of all ?
22 ADDITION. [SECT. II
32. What is the sum of 261+31+256+ 17 ?
33. What is the sum of 163+478+82+ 19 ?
34. What is the sum of 428+ 632+ 76+394 ?
35. W3iat is the sum of 320+ 856+100+503?36. What is the sum of 641+108+ 138+710 ?
37. What is the sum of 700+ 66+970+21 ?
38. What is the sum of 304+971+608+496 ?
39. What is the sum of 848+683+420+668 ?
40. What is the sum of 868+45+17+25+27+38?41. What is the sum of 641+85+580+42+7+63 ?
42. What is the sum of 29+281+7+43+785+46?43. A farmer sold 25 bushels of apples to one man, IT
bushels to another, 45 bushels to another, and 63 bushels
to another : how many bushels did he sell ?
44. A merchant bought one piece of cloth containing
25 yards, another 28 yards, another 34 yards, and an-
other 46 yards : how many yards did he buy ?
45. A man bought 3 farms ; one contained 120 acres,
another 246 acres, and the other 365 acres : how manyacres did they all contain ?
46. A traveler met four droves of cattle ; the first con-
tained 260, the second 175, the third 342, and the fourth
420 : hOw many cattle did the four droves contain ?
47. A carpenter built one house for 2365 dollars, an-
other for 1648 dollars, another for 3281 dollars, and an-
other for 5260 dollars : how much did he receive for all 9
48. Find the sum of six hundred and fifty-four, eighty-
nine, four hundred and sixty-three, and seventy-six.
49. Find the sum of two thousand and forty-seven,
three hundred and forty-five, thirty-six, and one hundred.
50. In January there are 31 days, February 28, March
81, April 30, May 31, June 30, July 31, August 31, .Sep-
tember 30, October 31, November 30, and December 31 :
how many days are there in a year ?
ART. 24.a.] ADDITION. 23
24ra. Accuracy and rapidity in adding can be ac-
quired only by practice. The following exercises are de-
signed to secure this important object.
OBS. 1. In solving the following examples, it is recommendedto the pupil simply to pronounce the result, as he adds each suc-
cessive figure. Thus, in Ex. 1, instead of saying 2 and 2 are 4,
and 2 are 6, &/c., proceed in the following manner :"two, four, six
;
eight, ten, twelve, fourteen, sixteen, eighteen, twenty." Set down
103. Multiply six hundred thousand, two hundred and
three by seventy thousand and seventeen.
QUEST. 48. When there are ciphers on tho right of both tl e inultipliel and
multiplicand, how proceed ?
ABTS. 48 50.] DIVISIDN. 49
SECTIt N V.
DIVISION.
ART. 49o Ex. 1. How many lead pencils, at 2 cents
apiece, can I buy for 1 cents ?
Solution. Since 2 cents will buy 1 pencil, 10 cents
will buy as many pencils, as 2 cents are contained times in
10 cents ;and 2 cents are contained in 10 cents, 5 times.
I can therefore buy 5 pencils.
2. A father bought 12 pears, which he divided equally
among his 3 children : how many pears did each re-
ceive ?
Solution. Reasoning in a similar manner as above, it
is plain that each child will receive 1 pear, as often as 3
is contained in 12 ; that is, each must receive as manypears, as 3 is contained times in 12. Now 3 is contained in
12, 4 times. Each child therefore received 4 pears.
OBS. The object of the first example is to find how many times
one given number is contained in another. The object of the second
is to divide a given number into several equal parts, and to ascertain
the value of these parts. The operation by which they are solved
is precisely the same, and is called Division. Hence,
5O. DIVISION is the process offinding how many times
one given number is contained in another.
The number to be divided, is called the dividend.
The number by which we divide, is called the divisor.
The ansiuer, or number obtained by division, is called
the quotient, and shows how many times the divisor is
contained in the dividend.
QUEST. 50. What is division ? What is the number to be divided, called ?
The number by which we divide ? What is the answer called? What does.ne quotient show ?
4
50 DIVISIO.V. [SECT. V
Note. The term quotient is derived from the Latin word qttotiea
which signifies how )ften, or how many times.
51. The number which is sometimes left after division,
in called the remainder. Thus, when we say 4 is con-
tamed in 21, 5 times and 1 over, 4 is the divisor, 21 the
dividend, 5 the quotient, and 1 the remainder.
OBS. 1. The remainder is always less than the divisor; for if it
were equal to, or greater than the divisor, the divisor could be con-
tained once more in the dividend.
2. The remainder is also of the same denomination as the divi-
dend; for it is a part of it.
52. Sign of Division (-r). The sign of Division is
a horizontal line between two dots (-7-), and shows that
the number before it, is to be divided by the number
after it. Thus, the expression 246, signifies that 24 is
to be divided by 6.
Division is also expressed by placing the divisor under
the dividend with a short line between them. Thus the
expressionA7*, shows that 35 is to be divided by 7, and is
equivalent to 35-7-7.
53* It will be perceived that division is similar in prin-
ciple to subtraction, and may be performed by it. For
instance, to find how many times 3 is contained in 12,
subtract 3 (the divisor) continually from 12 (the dividend)
until the latter is exhausted;then counting these repeated
subtractions, we shall have the true quotient. Thus, 3
from 12 leaves 9;3 from 9 leaves 6
;3 from 6 leaves 3
;
3 from 8 leaves 0. Now, by counting, we find that 3 has
QUEST. 51. What is the number called which is sometimes left after divi-
sion? When we say 4 is in 21, 5 times and 1 over, what is the 4 cabled? The21 ? The 5 ? The 1 V Obs. Is the remainder greater or less than the divisor?
Why? Of what denomination is it? Why? 52. What is the sign of divi-
aion ? What does it show ? In what other way is division expressed ?
ARTS. 51 53.] DIVISION. 51
been taken from 12, 4 times; consequently 3 is contained
t~ ?2, 4 times. Hence,
Division is sometimes defined to "be a short way of per-
forming repeated subtractions of the same number.
OBS. 1. It will also be observed that division is the reverse of
multiplication. Multiplication is the repeated addition of the same
number;
division is the repeated subtraction of the same number.
The product of the one answers to the dividend of the other : but
the latter is always given, while the former is required.
2. When the dividend denotes things of one kind, or denominar
tion only, the operation is called Simple Division.
DIVISION TABLE.
1 is in
52 DIVISION. [SECT. V
SHORT DIVISION.
ART. 54. Ex. 1. How many yards of cloth, at 2 dol
lars per yard, can I buy for 246 dollars ?
Analysis. Since 2 dollars will buy 1 yard, 246 dol-
lars will buy as many yards, as 2 dollars are contained
times in 246 dollars.
Directions. Write the divisor on Operation.
the left of the dividend with a curve w*>'- v
^n<L
line between them; then, beginning'
at the left hand, proceed thus: 2 isuot' l
contained in 2, once. As the 2 in the dividend denotes
hundreds, the 1 must be a hundred ; we therefore write
it in hundreds' place under the figure divided. 2 is con-
tained in 4, 2 times ; and since the 4 denotes tens, the 2
must also be tens, and must be written in tens' place. 2 is
in 6, 3 times. The 6 is units ; hence the 3 must be units,
and we write it in units' place. The answer is 123 yards.
Solve the following examples in a similar manner :
2. Divide 42 by 2. 6. Divide 684 by 2.
3. Divide 69 by 3. Y. Divide 4488 by 4.
4. Divide 488 by 4. 8. Divide 3963 by 3.
5. Divide 555 by 5. 9. Divide 6666 by 6.
55 When the divisor is not contained in the first
figure of the dividend, we must find how many times it
is contained hi the first two figures.
10. At 2 dollars a bushel, how much wheat can be
bought for 124 dollars?
Since the divisor 2, is not contained in Operation.
the first figure of the dividend, we find 2)124
how many times it is contained in the first Ans. 62 bu,
two figures. Thus 2 is in 12, 6 times ;set
the 6 under the 2. Next, 2 is in 4, 2 times. The an-
Ewer is 62 bushels.
ARTS 54 57.] DIVISION. 53
11. Divide 142 by 2. 13. Divide 1648 by 4.
12. Divide 129 by 3. 14. Divide 2877 by 7.
56 After dividing any figure of the dividend, if there
is a remainder, prefix it mentally to the next figure of the
dividend, and then divide this number as before.
Note. To prefix means to place before, or at the left hand.
15. A man bought 42 peaches, which he divided
equally among his 3 children : how many did he give to
each?
When we divide 4 by 3, there is 1 re- Operation,
mainder. This we prefix mentally to the 3)42
next figure of the dividend. We then say, 14 Ans.
3 is in 12, 4 times.
16. Divide 56 by 4. 18. Divide 456 by 6.
17. Divide 125 by 5. 19. Divide 3648 by 8.
57. Having obtained the first quotient figure, if the
divisor is not contained in any figure of the dividend, place
a cipher in the quotient, and prefix this figure to the next
one of the dividend, as if it were a remainder.
20. If hats are 2 dollars apiece, how many can be
bought for 216 dollars ?
As the divisor is not contained in 1, Operation.
the second figure of the dividend, we 2)216
put a in the quotient, and prefix the Ans. 108 hats.
1 to the 6 as directed above. Now 2
is in 16, 8 times.
21. Divide 2545 by 5. 23. Divide 6402 by 6.
22. Divide 3604 by 4. 24. Divide 4024 by 8.
25. A man divided 17 loaves of bread equally between
2 poor persons : how many did he give to each ?
Suggestion. Reasoning as before, he gave each as
many loaves as 2 is contained times in 17 :
54 DIVISION. [S2CT. V
Thus, 2 is contained in 17, 8 Opemtion-.
times and 1 over; that is, after 2)17
giving them 8 loaves apiece, there Quot. 8-1 remainder,
is one loaf left which is not divid- Ans. 8-J- loaves.
ed. Now 2 is not contained in 1;
hence the division must be represented by writing the 2
under the 1, thus , (Art. 52,) which jnust be annexed to
the 8. The true quotient, is 8-J. He therefore gave eight
and a half loaves to each. Hence,
58 When there is a remainder after dividing the last
figure of the dividend, it should always be written over the
divisor and annexed to the quotient.
Note. To annex means to place after, or at the riglit hand.
59* When the process of dividing is carried on in the
mind, and the quotient only is set down, the operation it
called SHORT DIVISION.
6O From the preceding illustrations and principles, wederive the following
RULE FOR SHORT DIVISION.
I. Write the divisor on the left of the dividend, with a
curve line between them.
Beginning at the left hand, divide each figure of the
dividend by the divisor, and place each quotient figure
under the figure divided.
II. When there is a remainder after dividing any fig-
ure, prefix it to the next figure of the dividend and divide
this number as before. If the divisor is not contained in
QUEST. 59. What is Short Division ? GO. How do you write numbersfor short division? Where begin to divide ? Where place each quotient fig-
ure? When there is a remainder after dividing a figure of the dividend,
what must be done with it 1 If the divisor is not contained in a fl ore of th*>
dividend, how proceed? When there is a remainder, after dividing the luat
fl#me of the dividend, what must be done with it ?
ARTS. 58 61.] DIVISION. 55
any figure of the dividend, place a cipher in the quotient,
and prefix this figure to the next one of the dividend, as if
it were a remainder. (Arts. 56, 57.)
III. When there is a ramainder after dividing the last
figurey write it over the divisor and annex it to the quotient,
61 PROOF. Multiply the divisor by the quotient, to
the product add the remainder, and if the sum is equal to
the dividend, the work is right.
OBS. Division may also be proved by subtracting the remainder,
if any, from the dividend, then dividing the result by the quotient.
EXAMPLES FOR PRACTICE.
1. Divide 426 by 3. 10. Divide 3640 by 5.
2. Divide 506 by 5. 11. Divide 6210 by 4.
3. Divide 304 by 4. 12. Divide 7031 by 7.
4. Divide 450 by 6. 13. Divide 2403 by 6.
5. Divide 720 by 7. 14. Divide 8131 by 9.
6. Divide 510 by 9. 15. Divide 7384 by 8.
7. Divide 604 by 5. 16. Divide 8560 by 7.
8. Divide 760 by 8. 17. Divide 7000 by 8.
9. Divide 813 by 7. 18. Divide 9100 by 9.
19. How many pair of shoes, at 2 dollars a pair, cau
you buy for 126 dollars ?
20. How many hats, at 4 dollars apiece, can be boughtfor 168 dollars?
21. A man bought 144 marbles which he divided equally
among his 6 children : how many did each receive ?
22. A man distributed 360 cents to a company of poor
children, giving 8 cents to each : how many children were
there in the company ?
23. How many yards of silk, at 6 shillings per yard,
can I buy for 450 shillings ?
QUEST. 61. How is division proved? Obs* What other wav of provingdivision is mentioned?
56 DIVISION. [SECT. V
24. A man having 600 dollars, wished to lay it out
in flour, at 7 dollars a barrel: how many whole barrels
could he buy, and how many dollars would he have left ?
25. If you read 9 pages each day, how long will it
take you to read a book through which has 828 pages?26. If I pay 8 dollars a yard for broadcloth, how many
yards can I buy for 1265 dollars?
27. If a stage coach goes at the rate of 8 miles per
hour, how long will it be in going 1560 miles ?
28. If a ship sails 9 miles an hour, how long will it
be in sailing to Liverpool, a distance of 3000 miles ?
LONG DIVISION.
ART. 62. Ex. 1. A man having 156 dollars laid it
out in sheep at 2 dollars apiece : how many did he buy ?
Analysis. Reasoning as before, since 2 dollars will
buy 1 sheep, 156 dollars will buy as many shee$ as 2
dollars are contained times in 156 dollars.
Directions. Having written the di- Operation.
visor on the left of the dividend as in Di - Divid -
^uot-
short division, proceed in the follow-34 .
ing manner :
First. Find how many times the-^
divisor (2) is contained in (15) the
first two figures of the dividend, and place the quotient
figure (7) on the right of the dividend with a curve line
between them. Second. Multiply the divisor by the
quotient figure, (2 times 7 are 14,) and write the product
(14) under the figures divided. Third. Subtract the
product from the figures divided. (The remainder is 1.)
Fourth. Bringing down the next figure of the dividend,
and placing it on the right of the remainder we have 16.
Now 2 is contained in 16, 8 times; place the 8 on the
right hand of the last quotient figure, and multiplying
ARTS. 62, G 3.] DIVISION. 57
the divisor by it, (8 times 2 are 16,) set the product undei
the figures divided, and subtract as before. Therefore 156
dollars will buy 78 sheep, at 2 dollars apiece.
63. When the result of each step in the operation is
set down, the process of dividing is called LONG DIVISION.
It is the same in principle as Short Division. The
only difference between them is, that in Long Division
the result of each step in the operation is written down,
while in Short Division we carry on the whole process
in the mind, simply writing down the quotient.
Note. To prevent mistakes, it is advisable to put a dot under
each figure of the dividend, when it is brought down.
Solve the following examples by Long Division :
2. Divide 195 by 3. Ans. 65.
3. Divide 256 by 2. 6, Divide 2665 by 5.
4. Divide 1456 by 4. V. Divide 4392 by 6.
5. Divide 5477 by 3. 8. Divide 6517 by Y.
OBS. When the divisor is not contained in the first two figures of
the dividend, find how many times it is contained in the first threct
or thefewest figures which will contain it, and proceed as before.
9. How many times is 13 contained in 10519?
Thus, 13 is contained in 105, Operation.
8 times; set the 8 in the quo- 13)l0519(809-t2srAns.
tient then multiplying and sub- 104
tracting, the remainder is 1. 119
Bringing down the next figure 117
we have 11 to be divided by 13. 2 rem.
But 13 is not contained in 11;
therefore we put a cipher in the quotient, and bring downthe next figure. (Art. 57.) Then 13 is sontained in 119,
CUTEST. 63. What is long division ? Wiiat is the diflerence between longed short division ?
58 DIVISION. [SECT "V.
9 times. Set the 9 in the quotient, multiply and sub-
tract, and the remainder is 2. Write the 2 over the di-
visor, and annex it to tho quotient. (Art. 58.)
O4. After the first quotient figure is obtained, for
each figure of the dividend which is brought down, either
a significant figure or a cipher must be put in the quotient.
Solve the following examples in a similar manner :
10. Divide 15425 by 11. Ans. 1402-ft-.
11. Divide 31237 by 15. Ans. 2082-ft.
65. From the preceding illustrations and principles
we derive the following
RULE FOR LONG DIVISION.
I. Beginning on the left of the dividend, find liow manytimes the divisor is contained in the fewest figures that will
contain it, and place the quotient figure on the right ofthe dividend with a curve line between them.
II. Multiply the divisor by this figure and subtract
the product from the figures divided ; to the right of
the remainder bring down the next figure of the dividend.
and divide this number as before. Proceed in this man-
ner till all the figures of the dividend are divided,
III. When there is a remainder after dividing the last
figure, write it over the divisor, and annex it to the quo-
tient, as in short division.
OBS. 1. Long Division is proved in the same manner as Short
Division.
2. When the divisor contains but one figure, the operation byShort Division is the most expeditious, and si ould therefore be
practiced; but when the divisor contains two or *r* \re figures, it will
generally be the most convenient to divide by Long Division.
QUEST. 65. How do you divide in long division? Where place the quo-tient ? Aftei obtaining the first quotient figure, how proceed ? When there is
a remainder after dividing the L'ist figure of the dividend what must be done
with it? Ols. How is long division proved? When should short division
be used ? Wheii long division ?
ARTS. 64, 65.] DIVISION. 59
EXAMPLES FOR PRACTICE.
1. Divide 369 by 8. 10. Divide 675 by 25.
2. Divide 435 by 9. 11. Divide 742 by 31.
3. Divide 564 by 7. 12. Divide 798 by 37.
4. Divide 403 by 10. 13. Divide 834 by 42.
5. Divide 641 by 11. 14. Divide 960 by 48.
6. Divide 576 by 12. 15. Divide 1142 by 53.
7. Divide 274 by 13. 16. Divide 2187 by 67.
8. Divide 449 by 14. 17. Divide 3400 by 75.
9. Divide 617 by 15. 18. Divide 4826 by 84.
19. How many caps, at 7 shillings apiece, can I buyfor 168 shillings?
20. How many pair of boots, at 5 dollars a pair, can
be bought for 175 dollars ?
<}1. A man laid out 252 dollars in beef, at 9 dollars a
barrel : how many barrels did he buy ?
22. In 12 pence there is 1 shilling : how many shillings
are there in 198 pence?23. In 20 shillings there is 1 pound : how many pounds
are there in 2 1 5 shillings ?
24. In 16 ounces there is 1 pound: how many poundsare there in 268 ounces ?
25. How many trunks, at 15 shillings apiece, can be
bought for 255 shillings ?
26. If 27 pounds of flour will last a family a week,
how long will 810 pounds last them?
27. How many yards of broadcloth, at 23 shillings per
yard, can be bought for 756 shillings?
28. If it takes 18 yards of silk to make a dress, how
many dresses can be made from 1350 yards?29. How many sheep, at 19 shillings per head, can be
bought for 1539 shillings?
30. A farmer having 1840 dollars, laid it out in land,
at 25 dollars per acre : how many acres did he buy?
60 DIVISION. [SECT. V
31. A man wishes to invest 2562 dollars in Railroad
stock : how many shares can he buy, at 42 dollars pershare ?
32. In 1 year there are 52 weeks: how many yearsare there in 1640 weeks ?
33. In one hogshead there are 63 gallons: how many
hogsheads are there in 3065 gallons ?
34. If a man can earn 75 dollars in a month, he wmanymonths will it take him to earn 3280 dollars ?
35. If a young man's expenses are 83 dollars a month,how long will 4265 dollars support him?
36. A man bought a drove of 95 horses for 4750 dol-olars : how much did he give apiece ?
37. If a man should spend 16 dollars a month, how
long will it take him to spend 172 dollars?
38. A garrison having 2790 pounds of meat, wished to
have it last them 3 1 days : how many pounds could theyeat per day ?
39. How many times is 54 contained in 3241, and how
many over ?
40. How many times is 68 contained in 7230, and how
many over ?
41. How many times is 39 contained in 1042, and how
many over?
42. How many times is 47 contained in 2002, and how
many over?
43. What is the quotient of 1704 divided by 56 ?
44. What is the quotient of 2040 divided by 60 ?
45. What is the quotient of 2600 divided by 49 ?
46. What is the quotient of 2847 divided by 81 ?
47. Divide 1926 by 75. 51. Divide 9423 by 105.
48. Divide 2230 by 85. 52. Divide 13263 by 112,
49. Divide 6243 by 96. 53. Divide 26850 by 123,
50. Divide 8461 by 99. 54. Divide 48451 by 224.
ARTS. 66, 67.] DIVISION. 6!
6G. It has been shown that annexing a cipher to a
number, increases its value ten times, or multiplies it by10, (Art. 44.) Reversing this process, that is, removing
a cipher from the right hand of a number, will evidently
diminish its value ten times, or divide it by 10; for, each
Sgure in the number is thus restored to its original place,
and consequently to its original value. Thus, annexing a
cipher to 12, it becomes 120, which is the same as 12 X 10.
On the other hand, removing the cipher from 120, it be-
comes 12, which is the same as 12010.In the same manner it may be shown, that removing
two ciphers from the right of a number, divides it by 100;
removing three, divides it by 1000; removing four, di-
vides It by 10000, &c. Hence,
67. To divide by 10, 100, 1000, &c.
Cut off as many figures from the right hand of the divi-
dend as there are ciphers in the divisor. The remaining
figures of the dividend will be the quotient, and those cut
off the remainder.
55. Divide 2456 by 100.
Since there are 2 ciphers on Operation.
the right of the divisor, we cut 1JOO)24|56off 2 figures on the right of the Quot. 24 and 56 rera
dividend. The quotient is 24
and 56 remainder, or 24-1V<r-
50. Divide 1325 by 10. Ans. 132 and 5 rem.
57. Divide 4620 by 100.
58. Divide 5633 by 1000.
59. Divide 8465 by 1000.
60. Divide 26244 by 1000.
61. Divide 136056 by 10000.
QuEST.~6<5. What is the effect of annexing a cipher to a number? What i*
the effect of removing a cipher from the right of a nr"aber ? 67, How proceedwben the divisor is 10, 100, 1000, &e.?
DIVISION* [SECT. V-
62. Divide 2443667 by 100000.
63. Divide 23454631 by 1000000.
68 When there are ciphers on the right hand of tho
divisor.
Cut off the ciphers from the divisor ; also cut of as
many figuresfrom the right of the dividend. Then divide
the remaining figures of the dividend by the remaining fig-ures of the divisor, and the result will be the quotient.
Finally, annex the figures cut offfrom the dividend to
the remainder, and the number thus formed will be the true
remainder.
64. At 200 dollars apiece, how many carriages can be
bought for 4765 dollars ?
Having cut off the two ciphers on
the right of the divisor, and two fig-
ures on the right of the dividend, we
divide the 47 by 2 in the usual way.
65. Divide 2658 by 20.
Ans. 132 and 18 rem
Operation.
2|OQ)47|65
Ans. 23 165 rera.
66. 3642 by 30.
68. 76235 by 1400.
70. 93600 by 2000.
72. 23148 by 1200.
74. 50382 by 1800.
76. 894000 by 2500.
78. 7450000 by 420000.
80. 348676 235.
82. 762005 401.
84. 6075071623.86. 4367238-7-2367.
88. 8230732-7-3478.
90. 93670S58-f-67213.
67. 6493 by 200
69. 82634 by 1600.
71. 14245 by 3000.
73. 42061 by 1500.
75. 88317 by 2100.
77. 9203010 by 3100.
79. 9000000 by 300000.
81. 467342 341.
83. 506725 603.
85. 736241 2764.
87. 6203451-7-3827.
89. 823o762-f-42316.
91. 98765421-7-84327.
QUEST. 68. When there are ciphers on the right of the divisor, how pro
ceed ? \Vhat is to be done with figures cut off fro in the dividend ?
ARTS. 68 73.] FRACTIONS. 63
SECTION VI.
FRACTIONS.
7 1 When a number or thing, as an apple or a pear,
is divided into two equal parts, one of these parts is called
one half. If divided into three equal parts, one of the
parts is called one third ; if divided into four equal parts,
one of the parts is called one fourth, or one quarter / if
into ten, tenths ; if into a hundred, hundredths, &Q.
When a number or thing- is divided into equal parts, as
halves, thirds, fourths, fifths, &c., these parts are called
Fractions. Hence,
72. A FRACTION denotes a part or parts of a number
or thing.
An Integer is a whole number.
Note. The term fraction, is derived from the Latin fractio,
which signifies the act of breaking, a broken part or piece. Hence,
fractions are sometimes called broken numbers.
73. Fractions are commonly expressed by two num-
bers, one placed over the other, with a line between them.
Thus, one half is written, -J-;one third, -J- ;
one fourth, J;three fourths, f- ;
two fifths, f ;nine tenths, -f^-, &c.
The number below the line is called the denominator,and shows into how many parts the number or thing is
divided.
QUEST. 71. What is meant by one half? What is meant by one third 7
What is meant by a fourth ? What are fourths sometimes called ? What is
meant by fifths ? By sixths ? Eighths ? How many sevenths mako a wholeone ? How many tenths ? What is meant by twentieths ? By hundrodths?72. What is a Fraction? What is an Integer? 73. How are fractions com-
monly expressed 1 What is the number below the line callud ? What does it
show ?
64 FRACTIONS. [SECT. VL
The number above the line is called the numerator, and
shows how many parts are expressed by the fraction.
Thus in the fraction-,the denominator .3, shows that the
number is divided into three equal parts ;the numerator 2,
shows that two of those parts are expressed by the fraction.
The numerator and denominator taken together, are
called the terms of the fraction.
7 4. A proper fraction is a fraction whose numerator is
less than its denominator; as, -|-, f, -f-.
An improper fraction is one whose numerator is equal
to, or greater than its denominator, as f, f-.
A simple fraction is a fraction which has but one nu-
merator and one denominator, and may be proper, or im-
proper ; as, f,
A compound fraction is a fraction of a fraction; as, J-
of
i of I.
A complex fraction is one which has a fraction in its
2i 4 24- -3-
numerator, or denominator, or in both;as
, -, f, ~.5 5 8f f
A mixed number is a whole number and a fraction writ-
ten together ; as, 4f, 25i.
7G The vato of a fraction is the quotient of the nu-
merator divided by the denominator. Thus, the value of
f is two ; of % is one ; of-J
is one third ; &c.
Read the following fractions, and name the kind of each :
1. f; f; f ; f; if; if; Y; W; W-2. -foff; f offof-V; * of of of 75.
QJ. 07 AI 3.
3. 2i; 14f ; 86;
QUEST. What is the number above the line called? What does it show?What are the denominator and numerator, taken together, culled ? 74. Whatis a proper fraction ? An improper traction V A simple fraction? A com-
pound fraction ? A complex fraction ? A mixed number ? 70. What is II*
value of a fraction?
ARTS. 74 78.] FRACTIONS.x
65
To find a fractional part of a given number.
Ex. 1. If a loaf of bread costs 4 cents, what will half a
loaf cost ?
Analysis. If a whole loaf costs 4 cents, 1 half a loaf
will cost 1 half of 4 cents;and 1 half of 4 cents is 2 cents
Half a loaf of bread will therefore cost 2 cents.
2. If a pound of sugar costs 12 cents, what will 1 third
of a pound cost ?
Analysis. Reasoning as before, if a whole pound costs
12 cents, 1 third of a pound will cost 1 third of 12 cents;
and 1 third of 12 cents is 4 cents. Ans. 4 cents.
77. From these examples the learner will perceive that
A half of a number is equal to as many units, as 2 is
contained times in that number.
A third of a number is equal to as many units as 3 is
contained times in that number.
A fourth of a numbei is equal to as many units, as 4 is
contained times in that number, &c. Hence,
78 To find a HALF of a number, divide it by 2.
To find a THIRD of a number, divide it by 3.
To find a FOURTH of a number, divide it by 4, &c.
Note. For mental exercises in Fractions, see Mental Arith-
metic, Section VII.
3. What is half of 257 ?
Dividing 257 by 2, the quotient is 128Operation.
and 1 over. Placing the 1 over the 22)257
and annexing it to the quotient, we ha\e 128i -Ans*
128-J-, which is the answer required.
4. What is a third of 21 ? 33 ? 48 ? 78 ! 151 !
5. What is a fourth of 45 ? 68 ? 72 ? 81 ? 130 I
6. What is a fifth of 7o ? 95 ? 135 ? 163 ?
7. What is an eighth of 73 ? 98 ? 104 ? 128 ?
QCKBT. 78. How do you find a half of a number ? A third ? A fonit t
3
66 FRACTIONS. [SECT. VI.
8. What is a seventh of 88 ? 133 ? 175 ? 250 ?
9. What is a ninth of 126 ? 163 ? 270 ? 316 ?
79 To find what part one given number is of another.
Make the number called the part, the numerator, and
the other given number the denominator. The fraction
thus formed will be the answer required.
1. What part of 3 is 2 ? Ans. f.
2. What part of 4 is 1? Is 2 ? Is 3 ? Is5?
3. What part of 7 is 2 ? Is 4 ? Is 5 ? Is 6 ?
4. What part of 9 is 1 ? Is 2 ? Is 4 ? Is 5?
5. 5 is what part of 11 ? Of 12 ? Of 13 ?
6. 8 is what part of 17 ? Of 19 ? Of 45 ?
7. 15 is what part of 38 ? Of 57 ? Of 85 ?
8O A part of a number being given to find the whole.
Multiply the given part by the number of parts into
which the whole is divided, and the product will be the
answer required.
1. 27 is 1 ninth of what number?
Suggestion. Since 27 is 1 ninth, 9 ninths, or the whole,
must be 9 times 27; and 27x9 243. Ans.
The given part is 27, and the Operation.
number of parts into which the 27= 1 ninth,
whole is divided, is 9 ninths; 9= no. parts,
we therefore multiply 27 by 9. Ans. 243= the whole.
2. 18 is 1 fifth of what number?3. 23 is 1 fourth of what number ?
4. 34 is 1 seventh of what number ?
5. 45 is 1 fifteenth of what number ?
6. 58 is 1 twelfth of what number?
7. 63 is 1 sixteenth of what number ?
QUEST. /9. How do you find what part one number is of another 1
80 When a part of a number is given, how do you find the whole ?
ARTS. 79 83.] FRACTIONS. 67
Multiplying a whole number by a fraction.
81. We have seen that multiplying by a whole num-ber is taking the multiplicand as many times as there are
units in the multiplier. (Art. 36.) On the other hand,
If the multiplier is only a part of a unit, it is plain wemust take only a part of the multiplicand. Hence,
82 Multiplying by a fraction is talcing a certain
PORTION of the multiplicand as many times as there are
like portions of a unit in the multiplier. That is,
Multiplying by -J-,is taking 1 half of the multiplicand
once. Thus, 6Xi 3.
Multiplying by --, is taking 1 third of the multiplicandonce. Thus, 6xi=2.
Multiplying by f, is taking 1 third of the multiplicandtwice. Thus, 6X1=4.
Obs. If the multiplier is a unit or 1, the product is equal to the
multiplicand ;if the multiplier is greater than a unit, the product
is greater than the multiplicand ; (Art. 36 j) and if the multiplier is
less than a unit, the product is less than the multiplicand. Hence,
83. To multiply a whole number by a fraction.
Divide the given number by tJie denominator, and mul-
tiply the quotient by the numerator.
Obs. 1. The result will be the same if vrefirst multiply the givennumber by the numerator, then divide this product by the denomi-
nator.
2. When the numerator is 1, it is unnecessary to multiply by it;
for, multiplying by 1 does not alter the value of a number. (Art.
82. Obs.)
QUEST. 81. What is meant by multiplying by a whole number? 82. By a
fraction? What is meant by multiplying by 1? By 4? By ? By |?Obs. If the multiplier is a unit or 1, what is the product equal to ? When the
multiplier is greater than 1, how is the product compared with the multipli-
cand ? When less, how ? 83. How do you multfply a whole number by a frac-
tion ? Obs. What other method is mentioned ? When flie muneiator ie 1, is
tt aece&sas y to multiply by it ? Why not ?
68 FRACTIONS. [SECT. VI,
Ex. 1. If a ton of coal costs 9 dollars, what -will a
ton cost ?
Suggestion. Since a whole ton costs Ojperc&i&ri.
9 dollars, 1 half a ton will cost 1 half of 2)9
9 dollars. Now 2 is contained in 9, 4 Ans. 4 dolls,
times and 1 over. Place the 1 over the
divisor 2, and annex it to the quotient. (Art. 58.)
2. What will f of a yard of cloth cost, at 36 shilling*
per yard ?
Suggestion. First find what 1 third First Operation.
of a yard will cost, then 2 thirds. 3)36
That is, divide the given number by 12
the denominator 3, then multiply the 2
quotient by the numerator 2. Ans. 24 shil.
Or, we may first multiply the given Second Operation
number by the numerator, then di- 36
vide the product by the denomina- 2
tor. The answer is the same as be- 3)72
fore. Ans. 24 shil.
3. If an acre of land costs 30 dollars, what will of
an acre cost ?
4. What will-J-
of a barrel of flour cost, at 40 shillings
per barrel ?
5. What will of a hogshead of molasses cost, at 37
dollars per hogshead ?
6. What will-f-
of a barrel of apples cost, at 28 shil-
fings per barrel ?
7.. Multiply 48 by f. 12. Multiply 56 by f.
8. Multiply 35 by f. 13. Multiply 72 by f.
9. Multiply 54 by i. 14. Multiply 120 by f.
10. Multiply 49 by f. 15. Multiply 168 by f.
11. Multiply 64 by -f. 16, Multiply 243 by \.
ABT 84.] FRACTIONS. 69
Multiplying a whole number by a mixed number.
17. What will 5 yards of cloth cost, at 18 shillings
per yard ?
Suggestion. Since 1 yard costs 2)18 cost of 1 yd.
18 shillings, 5- yards will cost 5- Si-
times as much. We first multiply 90 cost of 5 yds.
18 shillings by 5, then by , and 9 "of -J- yd.
add the products together. Hence, Ans. 99s. " of 5| yds.
84. To multiply a whole number by a mixed number
Multiply first by the whole number, then by the fraction,
and add the products together. (Art. 83.)
18. Multiply 26 by 2. Ans. 65.
19. Multiply 30 by 2-J-. 25. Multiply 75 by 2-f.
20. Multiply 36 by 3-J-. 26. Multiply 63 by 4-f.
21. Multiply 45 by 4. 27. Multiply 100 by 5f.
22. Multiply 42 by 5K 28. Multiply 165 by 7f.
23. Multiply 36 by 3*- 29. Multiply 180 by 8f.
24. Multiply 56 by H. 30. Multiply 192 by 9$.
31. Multiply 41 rods by 5-J-. Ans. 225 rods.
32. Multiply 68 rods by 16-J-.
33. What cost 21- acres of land, at 35 dollars pei
acre ?
34. What cost 34-J- hundred weight of indigo, at 47
dollars per hundred ?
35. What cost 63f tons of iron, at 96 dollars per ton?
Dividing a whole number by a fraction.
Ex. 1. How many apples at-J-
a cent apiece, can you
buy for 5 cents ?
Analysis. If % a cent will buy 1 apple, 5 cents will
buy as many apples, as-J- a cent is contained times in 5
cents ; that is, as many as there are halves in 5 whole ones,
QUEST. 84. How do you multiply a whole number by a mixed number ?
70 FRACTIONS. [SECT. VI
Now in 1 cent there are 2 halves, therefore in 5 cents
there are 5 times 2, which are 10 halves ; and 1 half is
containad in 10 halves, 10 times. Ans. 10 apples.
2. How many plums, at -f of a cent apiece, can you
buy for 8 cents ?
Analysis. Reasoning as before, you can buy as manyplums as | of a cent are contained times in 8 cents. Nowin 1 cent there are 3 thirds, therefore in 8 cents there are
8 times 3, which are 24 thirds, and 2 thirds are contained
in 24 thirds, 12 times. Ans. 12 plums. Hence,
8 5 To divide a whole number by a fraction.
Multiply the given number by the denominator, and
divide the result by the numerator.
OBS. When the numerator is 1, it is unnecessary to divide bjit
;for it is plain that dividing by 1 does not alter the value of e
number.
3. Divide 17 by i.
We multiply the 17 by the denominator Operation.
2; and since dividing by 1 does not alter 17
the value of a number, we do not divide 2
by it. 34 Ans
4. Divide 19 by -f.
Operation.
Multiply the 19 by 3, and divide the 19
product by 2. Place the remainder 1 3
over the divisor, and annex it to the 28. 2)57"
6. Divide 25 by -J-.10. Divide 89 by f.
6. Divide 38 by i. 11. Divide 123 by7. Divide 47 by \. 12. Divide 156 by8. Divide 63 by
-2V 13. Divide 190 by
-
9. Divide 72 by f. 14. Divide 256 by-
QUEST. 85. II ow do you divide a whole number by a fraction ? (Ift*.
the numerator is 1, is it necessary to divide by it? Why not?
ARTS. 85, 8G.] FRACTIONS. Yl
Dividing a whole number by a mixed number.
Ex. 1. How many lemons, at5-J-
cents apiece, can you
buy for 22 cents?
Analysis. Since 5j cents will buy 1 lemon, 22 cents
will buy as many lemons, as5-J- cents are contained times
in 22 cents. Now in 5|- cents there are 11 halves, and in
22 cents there are 44 halves;but 11 halves are contained
in 44 halves, 4 times. Ans. 4 lemons.
Suggestion. We change the divi-Operation,.
sor to halves by multiplying the whole51^22
number by the denominator 2, and 2 2
adding the numerator, we have 11 ji ^44halves
;then reducing the dividend to ^ns ^ jemong
halves by multiplying it by 2, we have
44 halves. Now 11 is contained in 44, 4 times. Hence,
86. To divide a whole number by a mixed number.
Multiply the whole number in the divisor by the denomi-
nator, and to the product add the numerator. Then mul-
tiply the dividend by the same denominator, and divide as
in ivhole numbers.
Note. For further illustrations of the principles of fractions see
Practical Arithmetic, Section VI. It is incompatible with the
design of the present work to treat of fractions more exten
sively than is necessary to enable the pupil to understand the
operations in Reduction.
EXAMPLES.
2. How many times is 4f- contained in 15 ?
Suggestion. Multiplying the 4 and 15 Qperationtby 3, reduces them to thirds. Now it is
.
plain we can divide thirds by thirds as 33well as we can divide one whole number
by another;
for the divisor is of the same_. "^ ~Q~T
name or denomination as the dividend.
QUEST. 86. How "lo you divide a whole number by a mixed number 7
72 FRACTIONS. [SECT. VI
3. Divide 18 by 1J. 7. Divide 46 by 7f.
4. Divide 20 by 3-J. 8. Divide 60 by 5i.
5. Divide 25 by 5f. 9. Divide 75 by 8f .
6. Divide 37 by 6-J-. 10. Divide 100 by lOf.
EXAMPLES FOB PRACTICE.
1. How many apples can you buy for 4 cents, if you
pay a cent apiece ?
2. How many peaches can you buy for 6 cents, if you
pay -J of a cent apiece ?
3. How many yards of tape can Sarah buy for 8 cents,
if she pays -J-of a cent a yard ?
4. How many yards of ribbon can Harriet buy for 9
shillings, at -f of a shilling per yard ?
5. How many pounds of tea, at f of a dollar a pound,can be bought for 6 dollars ?
6. How many yards of calico, at -J of a dollar per yaid,
can you buy for 3 dollars ?
7. At of a penny apiece, how many marbles can
George buy for 14 pence ?
8. At -f of a dollar a bushel, how many bushels of corn
can a man buy for 6 dollars ?
9. At -f of a dollar a yard, how many yards of silk can
a lady buy for 1 5 dollars ?
10. At -fc of a dollar apiece, how many lambs can a
drover buy for 27 dollars?
11. In 1 rod there are 5 yards : how many rods are
there in 88 yards ?
12. In 1 rod there are !<> feet: how many rods are
therein 132 feet?
13. How many yards of cloth, at 5f dollars per yard,
can be bought for 100 dollars?
14. How many cows, at 12 dollars apiece, can be
bought for 125 dollars ?
ART. 86.J
FRACTIONS. 73
15. How many acres of land, at 20-f- dollars per acre,
can a man buy for 540 dollars ?
16. A grocer bought a quantity of flour for 239 dol-
1 irs, which was 8- dollars per barrel : how many barrels
did he buy ?
17. A merchant bought a quantity of broadcloth, at
7f dollars per yard, and paid 372 dollars for it: how
many yards did he buy ?
18. A man hired a horse and chaise to take a ride, and
paid 275 cents for the use of it, which was 12-J- cents permile : how many miles did he ride ?
19. If a man hires a horse and carriage to go 1 Si-
miles, and pays 315 cents for it, how many cents does he
pay per mile ?
20. A young man hired himself out for 16f dollars per
month, and at the end of his time he received 201 dollars :
how many months did he work ?
21. A farmer having 261 dollars, wished to lay it out
in young cattle which were worth 10-f dollars per head :
how many could he buy ?
22. A man having 100 acres of land, wishes to find
how many building lots it will make, allowing -fa of an
acre to a lot : how many lots will it make ?
23. How many barrels of beef, at 9-J- dollars per barrel,
can be bought for 156 dollars ?
24. How many hours will it take a man to travel 250
miles, if he goes 12-J- miles per hour ?
25. In 1 barrel there are 31 gallons: how many bar-
rels are there in 315 gallons ?
26. A farmer paid 843 dollars for some colts, which
was 35^ dollars apiece : how many did he buy?27. A wagon maker sold a lot of wagons for 1452 dol-
lars, which was 45f dollars apiece : how many did he
sell?
74 COMPOUND [SECT. VIL
. SECTION VII.
COMPOUND NUMBERS.
ART. 87 SIMPLE Numbers are those which expressunits of the same kind or denomination ; as, one, two,three
;4 pears, 5 feet, &c.
COMPOUND Numbers are those which express units
of different kinds or denominations ; as the divisions of
money, weight, and measure. Thus, 6 shillings and 7
pence ;3 feet and 7 inches, &c., are compound numbers.
Note. Compound Numbers are sometimes called Denominate
Numbers.
FEDERAL MONEY.88. Federal Money is the currency of the United
States. Its denominations are, Eagles, dollars, dimes,
cents, and mills.
10 mills (m.) make 1 cent, marked ct.
10 cents "1 dime,
"d.
10 dimes "1 dollar,
" doll or $.
10 dollars"
1 eagle," E.
89 The national coins of the United States are of
three kinds, viz : gold, silver, and copper.
1. The gold coins are the eagle, half eagle, and quarter
eagle, the double eagle* and gold dollar.*
2. The silver coins are the dollar, half dollar, quarter
dollar, the dime, half dime, and three-cent-piece.
QUEST. 87. What are simple numbers ? What are compound numbers'?
88. What is Federal Money ? Recite the Table. 89. Of how many kinds are
the coins of the United States ? What are the gold coins ? What are the
silver coins ?
* Added by Act of Congress, Feb. 20th, 1849.
ARTS. 87 91.] NUMBERS. 75
3. The copper coins are the cent and half cent,
Mills are not coined.
Obs. Federal money was established by Congress, August 8th,
1786. Previous to this, English or Sterling money was the princi-
pal currency of the country.
STERLING MONEY.
90, English or Sterling Money is the national cur-
rency of Great Britain.
4 farthings (qr. or far.) make 1 penny, marked d.
12 pence" 1 shilling,
"s.
20 shillings" 1 pound or sovereign, .
21 shillings" 1 guinea.
OBS. The Pound Sterling is represented by a gold coin, called
a Sovereign. Its legal value, according to Act of Congress, 1842, is
$4.84; its intrinsic value, according to assays at the U. S. mint, is
$4.861. The legal value of an English shilling is 24-1 cents.
TROY WEIGHT.
91. Troy Weight is used in weighing gold, silver,
jewels, liquors, &c., and is generally adopted in philo-
sophical experiments.
24 grains (gr.) make 1 pennyweight, marked pwt.20 pennyweights
" 1 ounce,"
oz.
12 ounces "1 pound,
"Ib.
Note. Most children have very erroneous or indistinct ideas of
the weights and measures in common use. It is, therefore, stronglyrecommended for teachers to illustrate them practically, by referring
to some visible object of equal magnitude, or by 'exhibiting the ounce,the pound ;
the linear inch, foot, yard, and rod;also a square and
cubic inch, foot, &c.
QUEST. What are the copper coins ? Obs. When and by whom was Federal
Money established ? 90. What is Sterling Money ? Repeat the Table. Obs. Bywhat h the Pound Sterling represented ? What is its legal value in dollars andtents ? What is the value of an English shilling ? 91. in what is Troy Weightdeed ? Recite the Ttible,
*o COMPOUND [SECT. VII,
AVOIRDUPOIS WEIGHT.
92. Avoirdupois Weiylit is used in weighing groceries
and all coarse articles ;as sugar, tea, coffee, butter, cheese,
flour, hay, &c., and all metals except gold and silver.
16 drams (dr.) make 1 ounce, marked oz.
16 ounces "1 pound,
"Ib.
25 pounds"
1 quarter,"
qr.
4 quarters, or 100 Ibs."
1 hundred weight, ctvt.
20 hund., or 2000 Ibs."
1 ton, marked T.
OBS. 1. Gross weight is the weight of goods with the boxes, or bags
which contain them, allows 112 Ibs. for a hundred weight.Net weight is the weight of the goods only.
2. Formerly it was the custom to allow 112 pounds fora hundred
weight, and 28 pounds for a quarter : but this practice has become
nearly or quite obsolete. The laws of most of the states, as well as
general usage, call 100 Ibs. a hundred weight, and 25 Ibs. a quarter.In estimating duties, and weighing a few coarse articles, as iron,
dye-woods, and coal at the mines, 112 Ibs. are still allowed for a
hundred weight. Coal, however, is sold in cities, at 100 Ibs. for a
hundred weight.
APOTHECARIES' WEIGHT.
93. Apothecaries' Weight is used by apothecaries and
physicians in mixing medicines.
20 grains (yr.) make 1 scruple, marked sc. or S.
3 scruples"
1 dram,"
dr. or 3.
8 drams "1 ounce,
"oz. or g.
12 ounces "1 pound,
"Ib.
OBS. 1. The pound and ounce in this weight are the same as the
Troy pound and ounce; the subdivisions of the ounce are different.
2. Drugs and medicines are bought and sold by avoirdupois
weight.
QUEST. 92. In what is Avoirdupois Weight used ? Recite the Table. Obs
What is gross weight? What is net weight? How many pounds were for-
merly allowed for a quarter ? How many for a hundred weight ? 93. In what
is Apothecaries Weight used? Repeat the Table. Obs. To what are the Apo-thecaries' porn<l and o*ce equal? How are drugs and medicines boughtand sold ?
ARTS. 92 95.] NUMBERS. 77
LONG MEASURE.
O4r Long Measure is used in measuring length or
distances only, without regard to breadth or depth-
12 inches (in.)make 1 foot, marked ft.
3 feet" 1 yard,
"yd.
5i yards, or 16 feet"
1 rod, perch, or pole, r. orp40 rods " 1 furlong, marked fur.
8 furlongs, or 320 rods " 1 mile," m.
3 miles " 1 league," L
60 geographical miles, or )
691 statute.miles \"l d^ree-
" **'"*
360 deg. make a great circle, or the circum. of the eart li,
OBS. 1. 4 inches make a hand; 9 inches, 1 span; 18 inches, 1
cubit;6 feet, 1 fathom
;4 rods, 1 chain
;26 links, 1 rod.
2. Long measure is frequently called linear or lineal measure.
"Formerly the inch was divided into 3 barleycorns ; but the barley-
corn, as a measure, has become obsolete. The inch is commonlydivided either into eighths, or tenths ; sometimes it is divided into
twelfths, which are called lines.
CLOTH MEASURE.
95 Cloth Measure is used in measuring cloth, lace, and
all kinds of goods, which are bought or sold by the yard.
2J inches (in.) make 1 nail, marked na.
4 nails, or 9 in." 1 quarter of a yard,
"qr.
4 quarters" 1 yard,
"yd.
3 quarters" 1 Flemish ell,
" Fl. e.
5 quarters"
1 English ell," E. e.
6 quarters"
1 French ell," F. e.
QUEST. 94. In what is Long Measure used 1 Repeat the Table. Draw a
line an inch long upon your slate or black-board. Draw one two inches long.
Draw another a foot long. Draw one a yard long. How long is your teacher7!
desk 1 How long is the school-room ? How wide ? Obs. What is Long Meas-
ure frequently called 7 How is the inch commonly divided at the present
dav ? 95. In what is Cloth Measure used 1 Repeat the Table.
COMPOUND [SECT. VIL
OBS. Cloth mear ire is a species of long measure. The yard is
the eame in both. Cloths, laces, <fec., are bought and sold by the
linear yard, without regard to their width.
SQUARE MEASURE.
96* Square Measure is used in measuring surfaces,
or things whose length and breadth are considered with-
out regard to height or depth ; as land, flooring, plaster-
ing, &c.
144 square in. (sq. in.) make 1 square foot, marked sq.ft.
1 square yard, sq. yd.
1 sq. rod, perch, ((
ARTS. 96, 97.] NUMBERS. 79
1 cubic yard,"
80 COMPOUND [SECT. VIL
WINE MEASURE.98. Wine Measure is used in measuring wine, alco-
hol, molasses, oil, and all other liquids except beer, ale,
and milk.
4gills (gi.)
make 1 pint, marked pt.
2 pints" 1 quart,
"qt.
4 quarts" 1 gallon,
"gal.
3 1 gallons" 1 barrel,
" bar.orbbl.
42 gallons"
1 tierce,"
tier.
63 gallons, or 2 bbls."
1 hogshead," hkd.
2 hogsheads"
1 pipe or butt,"
pi.
2 pipes" 1 tun,
" tun,
OBS The wine gallon contains 231 cubic inches.
BEER MEASURE.99. Beer Measure is used in measuring beer, ale, and
milk.
2 pints (pt.) make 1 quart, marked qt.
4 quarts" 1 gallon,
"gal.
36 gallons"
1 barrel," bar. or bbl.
54 gals, or 1^ bbls."
1 hogshead," hhd.
OBS. The beer gallon contains 282 cubic inches. In many place*
milk is measured by wine measure.
DRY MEASURE.1 GO. Dry Measure is used in measuring grain, fruit
salt, &c.
2 pints (pts.) make 1 quart, marked qt.
8 quarts"
1 peck,"
pJc.
4 pecks, or 32 qts." 1 bushel,
" bu.
8 bushels "1 quarter,
"qr.
32 bushels " 1 chaldron,"
cA.
e.-t-ln England, 36 bushels of coal make a chaldron.
QUEST. 98. In what is Wine Measure used? Recite the Table. Obs. Ho*
many cubic inches in a wine gallon? 99. In what is Beer Measure uced
Repeat the Table. Obs. How many cubic inches in a boer gallon ?
ARTS. 98 102.] NUMBERS. 81
TIME.
1O1 Time is naturally divided into days and years ;
the former are caused by the revolution of the Earth on its
axis, the latter by its revolution round the Sun.
60 seconds(sec.)
make 1 minute, marked min.
60 minutes "1 hour,
"hr.
24 hours " 1 day,"
d.
7 days" 1 week,
" wk.
4 weeks "1 lunar month,
" mo.
12 calendar months, or > .. . .,'
x >" 1 civil year,
"yr.
365 clays, 6 hrs., (nearly,) $
13 lunar mo., or 52 weeks,"
1 year,"
yr.
100 years"
1 century,"
cen.
OBS. 1. Time is measured by clocks, watches, chronometers, dials,
hour-glasses, &c.
2. A civil year is a legal or common year ;a period of time es-
tablished by government for civil or common purposes.3. A solar year is the time in which the earth revolves round
the sun, and contains 365 days, 5 hours, 48 min., and 48 sec.
4. A leap year, sometimes called bissextile, contains 866 days,
and occurs once in four years.
It is caused by the excess of 6 hours, which the civil year con-
tains above 365 days, and is so called because it leaps or rims over
one day more than a common year. The odd day is added to Feb-
ruary, because it is the shortest month. Every leap year, there-
fore, February has 29 days.
1Q2. The names of the days are derived from the
names of certain Saxon deities, or objects of worship. Thus,
Sunday is named from the sun, because this day was dedicated
to its worship.
Monday is named from the moon, to which it was dedicated.
QUEST. 100. In what is Dry Measure used ? Recite the table. 101. Howis Time naturally divided ? How are the former caused ? How the latter t
Repeat the Table. Obs. How is Time measured 1 What is a civil year ? Asolar year? A leap year? How is Leap Year caused ? To which month is
the odd day added 1 From what are the namua of the days derived 1
6
82 COMPOUND [SECT. VII,
Tuesday is derived from Tuisco, the Saxon god of war.
Wednesday is derived from Woden, a deity of northern Europe.TJiursday is from Thor, the Danish god of thunder, storma, <fec.
Friday is from Friga, the Saxon goddess of beauty.
Saturday is from the planet Saturn, to which it was dedicated
1O3. The following are the names of the 12 calendar
months, with the number of days in each :
January,
February,
March,
April,
May,
June,
July,
August,
September,
October,
November,
December,
(Jan.) the first month, has 31 days.
(Feb.)" second " " 28 "
(Mar.)" third " "31 "
(Apr.) "fourth" " 30 "
(May)"
fifth" "31 "
(June)"
sixth " " 30 "
(July)" seventh " " 31 "
(Aug.)"
eighth" "31 "
(Sept.)" ninth " " 30 "
(Oct.)" tenth " "31 "
(Nov.)" eleventh " " 30 "
(Dec.)"
twelfth" "31 '
OBS. 1. The number of days in each month may be easily re-
membered from the following lines :
"Thirty days hath September,
April, June, and November;
February twenty-eight alone,
All the rest have thirty-one ;
Except in Leap Year, then is the time,
When February has twenty-nine."
2. The names of the calendar months were borrowed from the
Romans, and most of them had a fanciful origin. Thus,
January was named after Janus, a Roman deity, who WAS sup-
posed to preside over the year, and the commencement of all
undertakings.
February was derived fromfebrno, a Latin word which signifies
to purify by sacrifice, and was so called because this month was
devoted to the purification of the people.
QUEST. 103. What is the origin of the narr.es of the month*?
ARTS. 103 105.] NUMBERS. 88
March was named after Mars, the Roman god of war ;and was
originally the first month of the Roman year.
April, from the Latin aperio, to open, was so called from the
opening of buds, blossoms, cfec., at this season.
May was named after the goddess Maia, the mother of Mercury,to whom the ancients used to offer sacrifices on ike first day of
this month.
June was named after the goddess Juno, the wife of Jupiter.
July was so called in honor of Julius Ccesar, who was born in
this month.
August was so called in honor of Augustus Ccesar, a Roman
Emperor, who entered upon his first consulate in this month.
September, from the Latin numeral septem, seven, was so called,
because it was originally the seventh month of the Roman year.
It is the ninth month in our year.
October, from the Latin octo, eight, was so called because it was
the eighth month of the Roman year.
November, from the Latin novem, nine, was so called because it
was the ninth month of the Roman year.
December, from the Latin decem, ten, was so called because it
was the tenth month of the Roman year.
104. The year is also divided mtofour seasons of
three months each, viz: Spring, Summer, Autumn or
Fall, and Winter.
Spring comprises March, April, and May ; Summer,
June, July, and August ;Autumn or Fall, September,
October, and November ; Winter, December, Jan. and Feb.
CIRCULAR MEASyRE.
105. Circular Measure is applied to the divisions of
the circle, and is used in reckoning latitude and longitude,
and the motion of the heavenly bodies.
60 seconds (") make 1 minute, marked '
60 minutes " 1 degree,
30 degrees" 1 sign,
"s.
12 signs, or 360 "1 circle,
"c.
QUKST. 104. Name the seasons. 105. To what is Circular Measure applied 1
84 COMPOUND [SECT. VII.
OBS. J. Circular Measure is
c ten tailed Angular Measure,%nd is chiefly used by astrono-
mers, navigators, and surveyors.
2. The circumference of everycircle is divided, or supposed to
be divided, into 360 equal parts,
called degrees, as in the sub-
icined figure.
3. Since a degree is yfcr partof the circumference of a circle,
it is obvious that its lengthmust depend on the size of the circle.
270o
MISCELLANEOUS TABLE.
1O6. The following denominations not included in
the preceding Tables, are frequently used.
12 units
12 dozen, or 144
12 gross, or 1728
20 units
56 pounds100 pounds30 gallons
200 Ibs. of shad or salmon
196 pounds200 pounds14 pounds of iron, or lead
21 stone
8 pigs
OBS. Formerly 112 pounds were allowed for a quintal.
QUEST. Obs. What is Circular Measure sometimes called ? By whom ia i|
chiefly used ? Into what is the circumference of every circle divided ? Onwhat does the length of a degree depend ? 10G. How many units make a
dozen ? How many dozen a gross 1 A great gross ? How many units makaa score ? Pounds a flrjdn ?
make 1 dozen, (doz.)"
I gross."
1 great gross." 1 score.
"1 firkin of butter.
"1 quintal of fish.
"1 bar. of fish in Mass.
1 bar. in N. Y. and Ct1 bar. of flour.
1 bar. of pork.
1 stone.
1 P^1 fother.
ARTS. 106 108.] NUMBERS. 85
PAPER AND BOOKS.
1OT. The terms, folio, quarto, octavo, &c., applied to
books, denote the number of leaves into which a sheet
0f paper is folded.
24 sheets of paper make 1 quire.
20 quires"
1 ream.
2 reams " 1 bundle.
5 bundles "1 bale.
A sheet folded in two leaves, is called & folio.
A sheet folded in four leaves, is called a quarto, or 4te.
A sheet folded in eight leaves, is called an octavo, or 8vo.
A sheet folded in twelve leaves, is called a duodecimo.
A sheet folded in sixteen leaves, is called a 16wo.
A sheet folded in eighteen leaves, is called an 18wo.
A sheet folded in thirty-two leaves, is called a 39mo.
A sheet folded in thirty-six leaves, is called a 36wo.
A sheet folded in forty-eight leaves, is called a 48wo.
1O8 Previous to the adoption of Federal money in
1786, accounts hi the United States were kept in pounds,
shillings, pence, and farthings.
In New England currency, Virginia, Ken- i
tucky, Tennessee, Indiana, Illinois, Mis- >6 shil. make $1.
Bouri, and Mississippi, j
In New York currency, North Carolina, )8 shil makc &1 -
, )
]Ohio, and Michigan,In Pennsylvania currency, New Jersey, ) _,
Delaware, and Maryland, \7s ' 6d ' make $1
In Georgia currency, and South Carolina, 4s. 8d. make $1.
In Canada currency, and Nova Scotia, 5 shil. make $1.
QUEST. 107. When a sheet of paper is folded in two leaves, what is it
called ? When in four leaves, what ? When in eight ? In twelve ? In
sixteen 1 In eighteen 1 In thirty-six ? 108. Previous to the adoption of Fed-
eral Money, in what were accounts kept in the U. S. ? How many shillings
make a dollar in N. E. c\irrency 1 In N. V. currency ^ !\n Penn. currency 1
In Georgia currency 1 In Canada currency 7
86 COMPOUND [SECT. VIL
OBS. At the time Federal money was adopted, the colonial cur*
rency or bills of credit issued by the colonies, had more or less de-
preciated in value : that is, a colonial pound was worth less than a
pound Sterling; a colonial shilling, than a shilling Sterling, &e.
This depreciation being greater in some colonies than in others,
gave rise to the different values of the State currencies.
OBS. 1. In New York currency, it will be seen, (Art. 108,^ that
A six-pence, written 6d. = 6^ cents,
A shilling," Is. = 12 "
One (shil.) and 6 pence, 1/6. = 18J"
Two shillings,"
2s. = 25 "
PARTS OF $1 IN NEW ENGLAND CURRENCY.
3 shillings= $- 1 shilling
= $f2 shillings
= $-J-9 pence = $fc
I shil. and 6d. = $i 6 pence = $fV
OBS. 2. In New England currency, it will be seen, that
A four-pence-half-penny, written 4d. = 6 cents.
A six-pence," Gd. = 8 "
A nine-pence," 9d. = 12^
{C
A shilling,"
Is. = 16 f
One (shil.) and six-pence,"
1/6. = 25 "
Two shillings,"
2s. = 33 J"
QITEST. What are the aliquot parts of $1 in Federal Money 7 In New Yorkcurrency 7 In Now England currency 7 What are the aliquot parts of a pouSterling 7 Of a shilling 7
ART. 108.] NUMBERS. 87
ALIQUOT PARTS OF STERLING MONEY.
88 FEDERAL MONEY. [SECT. VIIL
SECTION VIIL
FEDERAL MONEY.
1 1 0. Accounts in the United States are kept in dol-
lars, cents, and mills. Eagles are expressed in dollars, and
dimes in cents. Thus, instead of 4 eagles, we say, 40 dol-
lars;instead of 5 dimes, we say, 50 cents, &c.
Ill* Dollars are separated from cents by placing a
point or separatrix (.) between them. Hence,
112. To read Federal Money.
Call all the figures on the left of the point, dollars ; the
first two figures on the right of the point, are cents ; the
third figure denotes mills ; the other places on the right are
parts of a mill. Thus, $5.2523 is read, 5 dollars, 25
cents, 2 mills, and 3 tenths of a mill.
OBS. 1. Since two places are assigned to cents, when no cents
are mentioned in the given number, two ciphers must be placedbefore the mills. Thus, 5 dollars and 7 mills are written $ 5.007.
2. If the given number of cents is less than ten, a cipher must
always be written before them. Thus, 8 cents are written .08, <fcc.
1. Read the following expressions: $83.635 ; $75.50.
$126.607; $268.05; $382.005; $2160.
2. Write the following sums : Sixty dollars and fifty
cents. Seventy-five dollars, eight cents, and three mills.
Forty-eight dollars and seven mills. Nine cents. Six
cents and four mills.
QUEST. 88. What is Federal Money? What are its denominations? Re-
cite the Table. 110. How are accounts kept iu the United States? How are
Eagles expressed ? Dimes? 111. How are dollars distinguished from cents
and mills ? 112. How do you read Federal Money ? Obs. How many places
are aseigned to cents ? When the number of cents is less than ten, what must
be done ? When no cents are mentioned, what do you do ?
ARTS. 110 113.] FEDERAL MONEY. 89
REDUCTION OF FEDERAL MONEY.
CASE I.
Ex. 1, How many cents are there in 65 dollars?
Suggestion. Since in 1 dollar there are Operation.
100 cents, in 65 dollars there are 65 times as 65
many. Now, to multiply by 10, 100, &c.,we
annex as many ciphers to the multiplicand,6500 cents.
as there are ciphers in the multiplier. (Art. 45.) Hence,
113. To reduce dollars to cents, annex TWO ciphers.
To reduce dollars tc mills, annex THREE ciphers.
To reduce cents to mills, annex ONE cipher.
OBS. To reduce dollars and cents to cents., erase the sign of dollars
and the separatrix. Tfefes, $25.36 reduced te cents, become 2536
cents.
2. Reduce $4 to cents. Ans. 400 cents.
3.* Reduce $15 to cents. 7. Reduce $96 to mills.
4. Reduce $27 to cents. 8. Reduce $12.23 to cents.
5. Reduce $85 to cents. 9'. Reduce $86.86 to cents.
6. Reduce $93 to cents. 10. Reduce $9.437 to mills.
CASE II.
1. In 2345 cents, how many dollars ?
Suggestion. Since 100 cents make 1 dol- Operation.
lar, 2345 cents, will make as many dollars 1|00)23|45
as 100 is contained times in 2345. Now to Ans. $23.45
divide by 10, 100, &c., we cut off as many
figures from the right of the dividend as there are ciphera
in the divisor. (Art. 67.) Hence,
QUEST. 113. Ho\v are dollars reduced to cents? Dollars to mills ? Centa
to mills ? Obs. Dollars and cents to cents ?
90 FEDERAL MONET. [SECT. VIIL
114* To reduce cents to dollars.
Point off TWO figures on the right ; the figures remain*
ing on the left express dollars ; the two pointed off, cents.
1 1 5. To reduce mills to dollars.
Point off THREE figures on the right ; the remaining
figures express dollars ; the first two on the right of the
point, cents ; the third one, mills.
116* To reduce mills to cents.
Point off ONE figure on the right, and the remaining
figures express cents ; the one pointed off, mills.
2. Reduce 236 cts. to dolls. 3. Reduce 21 63 cts. to dolls.
4. Reduce 865 mills to dolls. 5. Reduce 906 mills to cts.
6. Reduce 2652 cts. to dolls. 7. Redfce 3068 cts. to dolls.
ADDITION OF FEDERAL MONEY.
Ex. 1. What is the sum of $8.125, $12.67, $3.098, $11 ?
Suggestion. Write the dollars under
dollars, cents under cents, mills tinder 1267mills, and proceed as in Simple Addition. 3 098From the right of the amount point off 11.00
three figures for cents and mills. Ans. $34.893
117* Hence, we derive the following general
RULE FOR ADDING FEDERAL MONEY.
Write dollars under dollars, cents under cents, mills
under mills, and add each column, as in simple numbers.
From the right of the amount, point off as many figures
for cents and mills, as there are places of cents and mills
in either of the given numbers.
QUEST.- 114. How are cents reduced to dollars? 115. Mills to dollars? 117
How do you add Federal Money? How point off the amount?
ARTS. 114 11*7.] FEDERAL MONEY. 91
OBS. If either of the given numbers have no cents expressed,
upply their place by ciphers.
(2.)
$375.037
92 FEDERAL MONEY. [SECT. VIIL
SUBTRACTION OF FEDERAL MONEY.Ex. 1. What is the difference between $845.634, and
$86.087 ?
Suggestion. Write the less number Operation.
under the greater, dollars under dollars, $845.634
&c., then subtract, and point off the an- 86.087
swer as in addition of Federal Money. Ans. $759.547.
118* Hence, we derive the following general
EULE FOR SUBTRACTING FEDERAL MONEY.
Write the less number under the greater, with dollars
under dollars, cents under cents, and mills under mills ;
then subtract, and point off the answer as in addition
of Federal Money.
(2.)
From $856.453
Take $387.602
ARTS. 118, 119.] FEDERAL MONEY. 93
MULTIPLICATION OF FEDERAL MONEY.Ex. 1. What will 3 caps cost, at $1.625 apiece ?
Suggestion. Since 1 cap costs $1.625,Operation.
8 caps will cost 3 times as much. We *^ ^25therefore multiply the price of 1 cap by 3, 3
the number of caps, and point off three Ans. $4.875.
places for cents and mills. Hence,
111). When the multiplier is a whole number, we have
the following
RULE FOR MULTIPLYING FEDERAL MONEY.
Multiply as in simple numbers, andfrom the right ofthe product, point off as many figures for cents and mills,
as there are places ofcents and mills in the multiplicand.
OBS. 1. In Multiplication of Federal Money, as well as in simple
numbers, the multiplier must always be considered an abstract
number.
2. In business operations, when the mills are 5 or over, it is
customary to call them a cent;when under 5, they are disregarded.
(2.) (3.) (4.) (5.)
Multiply $633.75 $805.625 $879.075 $9071.26
By 8 9 24 37
(6.)
94 FEDERAL MONEY. [SECT. VIIL
15. At $2.63 apiece, what will 15 chairs come tof
16. What costs 25 Arithmetics, at 37-^ cents apiece!
17. "What cost 38 Readers, at 62|- cents apiece?
18. What cost 46 over-coats, at $25.68 apiece ?
19. What cost 69 oxen, at $48.50 a head ?
20. At $23 per acre, what cost 65 acres of land ?
21. At $75.68 apiece, what will 56 horses come to ?
22. At 7-J cents a mile, what will it cost to ride 100
miles ?
23. A farmer sold 84 bushels of apples, at 87-J- cents per
bushel : what did they come to 1
24. If I pay $5.3 7|- per week for board, how much will
it cost to board 52 weeks ?
DIVISION OF FEDERAL MONEY.
Ex. 1. If you paid $18.876 for 3 barrels of flour, how
much was that a barrel ?
Suggestion. Since 3 barrels cost $18.-
876, 1 barrel will cost 1 third as much,
We therefore divide as in simple division,
and point on three places for cents and
mills, because there are three in the dividend. Hence,
1 2O. When the divisor is a whole number, we have
the following
RULE FOR DIVIDING FEDERAL MONEY.
Divide as in simple numbers, andfrom the right of the
quotient, point off as many figures for cents and mills, as
there are places of cents and mills in the dividend.
OBS. "When the dividend contains no cents and mills, if there
is a remainder annex three ciphers to it;then divide a& before,
and point off three figures in the quotient.
QUKST. 320. How do you divide Federal Money ? How point off the
quotient ? Obs> When the dividend contains no cents and mills, how f roceed ?
AAT. 120.] FEDERAL MONEY 95
Note. For a more complete development of multiplication and
division of Federal Money, the learner is referred to the author's
Practical and Higher Arithmetics.
When the multiplier or divisor contain decimals, or cents and
mills, to understand the operation fully, requires a thorough
knowledge of Decimal Fractions, a subject which the limits of this
work will not allow us to introduce,
(2.) (3.) (4.)
6) $856.272. 8) $9567.648. 9) $7254,108.
5. Divide $868.36 by 27. 6. Divide $3674.65 by 38.
7. Divide $486745 by 49. 8. Divide $634.075 by ofi.
9. Divide $6634.25 by 60. 10. Divide $5340.73 by 78
11. Divide $7643.85 by 83. 12. Divide $4389.75 by 89.
13. Divide $836847 by 94. 14. Divide $94321.62 by 97.
15. A man paid $2563.84 for 63 sofas : what was that
apiece ?
16. A miller sold 86 barrels of flour for $526.50 : howmuch was that per barrel ?
17. If a man pays $475.56 for 65 barrels of pork, what
is that per barrel ?
18. A man paid $1875.68 for 93 stoves: how muchwas that apiece ?
19. If $2682.56 are equally divided among 100 men,
how much will each receive ?
20. A cabinet-maker sold 116 tables for $968.75 : howmuch did he get apiece ?
21. A farmer sold 168 sheep for $465 : how much did
he receive apiece for them ?
22. A miller bought 216 bushels of wheat for $375.50 :
how much did he pay per bushel ?
23. If $2368.875 were equally divided among 348 per-
sons, how much would each person receive ?
98 REDUCTION. [SECT. IX.
SECTION IX
REDUCTION.
ART. 121* REDUCTION is the process of changing
Compound Numbers from one denomination into another
without altering their value.
REDUCING HIGHER DENOMINATIONS TO LOWER.
122. Ex. 1. Reduce 2, to farthings.
Suggestion. First reduce theOperation.
given pounds (2) to shillings, by 2
multiplying them by 20, because 20s. in l.
20s. make l. Next reduce the 40 shillings,
shillings (40) to pence, by multi- I2d. in Is.
plying them by 12, because 12d. 480 pence.
make Is. Reduce the pence (480) _1far- in ld-
to farthings, by multiplying them ^ns- 192 farthings.
by 4, because 4 far. make Id.
2. Reduce l, 2s. 4d. and 3 far. to farthings.
Suggestion. In this example Operation.
there are shillings, pence, and far- * d. far.
things. Hence, when the pounds* 2
.
4 3
are reduced to shillings, the given
shillings (2) must be added men-itd^iJT
tally to the product. When the' x, , ,
268 pence,shillings are reduced to pence, the
4 f -'
id
given pence (4) must be added; ^ ^^^ '
and when the pence are reduced to
farthings, the given farthings (3) must be added.
Q,UEST. 121. What is reduction? 122. Ex. 1. How reduce pounds to shil-
lings? Why multiply by 20 ? How are shillings reduced to pence? Why ?
How pence to farthings 1 Why ?
ARTS. 121 124.] REDUCTION. 0Y
OBS. lu these examples it is required to reduce higher denomi
nations to lower, as pounds to shillings, shillings to pence, <fcc.
123* The process of reducing higher denominationa
to lower, is usually called Reduction Descending.
It consists in successive multiplications, and may with
propriety be called Reduction by Multiplication.
124* From the preceding illustrations we derive the
following
RULE FOB REDUCTION DESCENDING.
Multiply the highest denomination given by the num-
ber required of the next lower denomination to make ONE
of this higher, and to the product, add the given number ofthis lower denomination.
Proceed in this manner with each successive denomina-
tion, till you come to the one required.
EXAMPLES.
3. Reduce 4 miles, 2 fur., 8 rods and 4 feet to feet.
Operation.
Suggestion. Having reduced the m.fur. r. ft.
miles and furlongs to rods, we have
1368 rods. We then multiply by .
10-J-, because 16^- feet make 1 rod.,Q
(Art. 94.) Now 16J is a mixed2)1368 rods,
number; we therefore first multi- jgi
ply by the whole number (16), 8212then by the fraction (-), and add 1368the products together. (Art. 84.) 684
Ans. 22576 feet.
QUEST. 123. What is reducing compound numbers to lower denominations
usually called? Which of the fundamental rules is employed in reduction
descending? 124. What is the rule for Reduction Defending ?
98 REDUCTION. |SECT. IX
4. In 5, 16s. 7d., how many farthings ? Ans. 5596 far
5. In 18 how many pence?6. In 23, 9s., how many shillings ?
13. In 13 Ibs. 4 oz. avoirdupois, how many ounces?
14. In 2 qrs. 17 Ibs. avoirdupois, how many pounds f
15. In 6 Ibs. 12 oz. avoirdupois, how many drams?
16. In 12 cwt. 1 qr. 6 Ibs. avoir., how many ounces *
17. In 16 miles, how many rods?
18. In 28 rods and 2 feet, how many inches ?
19. In 19 fur. 4 rods and 2 yds., how many feet ?
20. In 25 leagues and 2 m., how many rods ?
21. Reduce 14 yards cloth measure to quarters.
22. Reduce 21 yards 2 quarters to nails.
23. Reduce 17 yards 3 quarters 2 nails, to nails.
24. How many quarts in 23 gallons, wine measure ?
25. How many gills in 30 gallons and 2 quarts?26. How many gills in 63 gallons ?
27. How many quarts in 41 hogshead* ?
28. How many pecks in 45 bushels ?
29. How many pints in 3 pecks and 2 quarts ?
30. How many quarts in 52 bu. and 2 peck* ?
31. How many hours in 15 weeks?
32. How many minutes in 25 days ?
83. How many seconds in 265 hours ?
34. How many minutes in 52 weeks ?
35. How many seconds in 68 days ?
ARTS. 125, 126.] REDUCTION. 99
REDUCING LOWER DENOMINATIONS TO HIGHER.
125. Ex. 1. Reduce 1920 farthings to pounds.
Suggestion. First reduce the given far- Operation.
things (1920) to pence, the next higher 4)1920 far,
denomination, by dividing them by 4, be- 12)480d.cause 4 far. make Id. Next reduce the 20)40s.
pence (480) to shillings, by dividing them 2 Ans.
by 12, because 12d. make Is. Finally, re-
duce the shillings (40) to pounds, by dividing them by 20,
because 20s. make l. The answer is 2. That is, 1920
far. are equal to 2.
2. In 1075 farthings, how many pounds?
Suggestion. In dividing the Operation.
given farthings by 4, there is a 4)1075 far.
remainder of 3 far., which should 12)268d. 3 far. over,
be placed on the right. In di- 20)22s. 4d. over,
viding the pence (268) by 12, 1, 2s. over,
there is a remainder of 4d., which Ans. l, 2s. 4d. 3 far.
should also be placed on the
right. In dividing the shillings (22) by 20, the quotient
is l and 2s. over. The last quotient with the several
remainders is the answer. That is, 1075 far. are equal to
1, 2s. 4d. 3 far.
OBS. In the last two examples, it is required to reduce lower de-
nominations to higher, as farthings to pence, pence to shillings, &c.
The operation is exactly the reverse of that in Reduction Descending.
126* The process of reducing lower denominations to
higher is usually called Reduction Ascending.
It consists in successive divisions, and may with propri-
ety be called Reduction by Division.
QUFST. 125. Ex. 1. How are farihings reduced to pence ? Why divide by 4 1
How reduce pence to shillings 1 WLy? How shillings to pounds? Why?120. What is reducing compound numbers to higher denominations usually
called' Which ol the fundamental rules ia employed in Reduction Ascending ?
100 REDUCTION. [SECT. IX.
127. From the preceding illustrations we derive the
following
RULE FOR REDUCTION ASCENDING.
Divide the given denomination ~by that number which it
takes of this denomination to mafee ONE of the next higher.
Proceed in this manner with each successive denomination)
till you come to the one required. The last quotient, with
tlie several remainders, will be the answer sought.
128* PROOF. Reverse the operation; that is, reduce
back tlie answer to the original denominations, and if the
result correspond with the numbers given, the work is right.
OBS. Each remainder is of the satne denomination as the divi-
dend from which it arose. (Art. 51, Obs. 2.)
EXAMPLES.
3. In 429 feet, how many rods ? Operation,
Suggestion. We first reduce the feet 3 )429 feet.
to yards, then reduce the yards to rods 5i-)143 yds.
by dividing them by 5-J-. (Art. 86.) 2 2
Or, we may divide the given feet by 11 )286
16, the number of feet in a rod, and the Ans. 26 rods,
quotient will be the answer.
Proof.
We first reduce the rods back to yards, 26 rods.
(Art. 84,) then reduce the yards to feet. 5^The result is 429 feet, which is the same 130
as the given number of feet. 13
Or, we may multiply the 26 by 16i, 143 yds.
and the product will be 429. 3
429 feet.
4. Reduce 256 pence to pounds. Ans. l, Is. 4d.
5. Reduce 324 pence to shillings.
QUEST. 127, What is the rule for reduction ascending ? 328, Hovr is re-
duction proved 1 Obs. Of what denomination is each remainder 7
ARTS. 127 129.] REDUCTION. 101
6. Reduce 960 farthings to shillings.
7. Reduce 1250 farthings to pounds.8. In 265 ounces Troy weight, how many pounds ?
9. In 728 pwts., how many pounds Troy? %
10. In 54'8 grains, how many ounces Troy?11. In 638 oz. avoirdupois weight, how many pounds?12. In 736 Ibs. avoirdupois, how many quarters?13. In 1675 oz. avoirdupois, how many hundred weight ?
14. In 1000 drams avoirdupois, how many pounds?15. In 4000 Ibs. avoirdupois, how many *,ons?
16. How many yards in 865 inches ?
17. How many rods in 1000 feet?
18. How many miles in 2560 rods ?
19. How many miles in 3261 yards ?
20. How many leagues in 2365 rods ?
EXAMPLES IN REDUCTION ASCENDING AND DESCENDING.
129* In solving the following examples, the pupil
tnust first consider whether the question requires higher
denominations to be reduced to lower, or lower denomina-
tions to higher. Having settled this point, he "ill find no
difficulty in applying the proper rule.
FEDERAL MONEY. (ART. 88.)
1. In 3 dollars and 16 cents, how many cents ?
2. In 81 cents and 2 mills, how many mills?
3. In 245 cents, how many dollars?
4. In 321 mills, how many dimes?
5. In 95 eagles, how many cents ?
6. In 160 dollars, how many cents ?
7. In 317 dollars, how many dimes?
8. In 4561 mills, how many dollars?
9. In 8250 cents, how many eagles ?
10. In 61 dolls., 12 cts., *md 3 mills, how many milk?
02 REDLCTION. [SECT. IX.
STERL .NG MONEY. (ART. 90.)
11. Keduce 17, 16s. to shillings.
12: Reduce 19s. 6d. 2 far. to farthings.
13. ^Reduce 1200 pence to pounds.14. Reduce 3626 farthings to shillings.
15. Reduce 19 to farthings.
16. Reduce 2880 farthings to shillings.
17. Reduce 21, 3s. 6d. to pence.18. Reduce 3721 farthings to pounds.
TROY WEIGHT. (ART. 91.)
.19. In 7 Ibs., how many ounces ?
20. In 9 Ibs. 2 oz., how many pennyweights ?
21. In 165 oz., how many pounds?22. In 840 grains, how many ounces ?
23. In 3 Ibs. 5 oz. 2 pwts. 7 grs., how many grains?24. In 6860 grains, how many pounds?
AVOIRDUPOIS WEIGHT. (ART. 92.)
25. In 200 oz., how many pounds ?
26. In 261 Ibs., how many ounces?
27. In 3 tons, 2 cwt., how many pounds?28. In 1 cwt. 2 qrs., how many ounces ?
29. In 1000 oz., how many pounds?30. In 4256 Ibs., how many tons ?
APOTHECARIES' WEIGHT. (ART. W.)
31. Reduce 45 pounds to ounces.
32. Reduce 71 oz. to scruples.
33. Reduce 93 Ibs. 2 oz. to grains.
34. Reduce 165 oz. to pounds.
35. Reduce 962 drams to pounds.
LONG MEASURE. (ART. 94 )
36. In 636 inches, how many yards ?
87. In 763 feet, how many rods?
ART. 129 ] REDUCTION'. 103
38. In 4 miles, how many feet ?
39. In 18 rods 2 feet, how many inches ?
40. In 1760 yards, how many miles?
41. In 3 leagues, 2 miles, how many inches?
CLOTH MEASURE. (ART. 95.)
42. How many yards in 19 quarters?43. How many quarters in 21 yards and 3 quaiters?44. How many nails in 35 yards and 2 quarters?45. How many Flemish ells in 50 yards ?
46. How many English ells in 50 yards ?
47. How many French ells in 50 yards ?
SaUARE MEASURE. (ART. 96.)
48. In 65 sq. yards and 7 feet, how many feet ?
49. In 39 sq. rods and 15 yds., how many yards?50. In 27 acres, how many square feet?
51. In 345 sq. rods, how many acres ?
52. In 461 square yards, how many rods?
53. In 876 sq. inches, how many sq. feet ?
CUBIC MEASURE. (ART. 97.)
54. In 48 cubic yards, how many feet ?
55. In 54 cubic feet, how many inches ?
56. In 26 cords, how many cubic feet ?
57. In 4230 cubic inches, how many feet?
58. In 3264 cubic feet, how many cords ?
WINE MEASURE. (ART. 98.)
59. Reduce 94 gallons 2 qts. to pints.
60. Reduce 68 gallons 3 qts. to gills.
61. Reduce 10 hhds. 15 gallons to quarts.
62. Reduce 764 gills to gallons.
63. Reduce 948 quarts to hogsheads.64. Reduce 896 gills to gallons.
J V4 REDUCTION. [SECT. IX.
BEER MEASURE. (ART. 99.)
65. How many quarts in 1 1 hogsheads of beer ?
66. How many pints in 110 gallons 2 qts. of beer ?
67. How many hogsheads in 256 gallons of beer?
68. How many barrels in 320 pints of beer?
69. How many pints in 46 hhds. 10 gallons ?
70. How many hhds. in 2592 quarts ?
DRY MEASURE. (ART. 100.)
71. In 156 pecks, how many bushels ?
72. In 238 quarts, how many bushels ?
73. In 360 pints, how many pecks ?
74. In 58 bushels, 3 pecks, how many pecks ?
75. In 95 pecks, 2 quarts, how many quarts ?
76. In 373 quarts, how many bushels ?
77. In 100 bushels, 2 pecks, how many pints?
TIME. (ART. 101.)
78. How many minutes in 16 hours ?
79. How many seconds in 1 day?80. How many minutes in 365 days ?
81. How many days in 96 hours ?
82. How many days in 3656 minutes ?
83. How many seconds in 1 week ?
84. How many years in 460 weeks?
CIRCULAR MEASURE. (ART. 105.)
85. Reduce 23 degrees, 30 minutes to minutes.
86. Reduce 41 degrees to seconds.
87. Reduce 840 minutes to degrees.
88. Reduce 964 minutes to signs.
89. Reduce 2 signs to seconds.
90. Reduce 5 signs, 2 degrees to minutes.
91. Reduce 960 seconds to degrees.
92. Reduce 1800 minutes to signs.
ART. 117.] REDUCTION. 105
93. In 45 guineas, how many farthings \
94. In 60 guineas, how many pounds ?
95. In 62564 pence, how many guineas ?
96. In 84, how many guineas ?
97. How many grains Troy, in 46 Ibs. 7 oz. 5 pwts. ?
98. How many pounds Troy, in 825630 grains ?
99. Reduce 62 Ibs. 10 pwts. to grains.
100. In 16 tons, 11 cwt. 9 Ibs., avoir., how many pounds ?
101. Reduce 782568 ounces to tons.
102. In 18 rods, 2 yds. 3 ft. 10 in., how many inches ^
103. How many feet in 3 leagues, 2 miles, 12 rods ?
104. In 2738 inches, how many rods ?
105. In 2 tons, 3 cwt. 2 qrs. 15 Ibs., how many ounces !